Find the Volume of Each Pyramid. 1. SOLUTION

1. SOLUTION:

ANSWER: 12-5 Volumes of Pyramids and Cones 3 75 in

Find the volume of each pyramid.

1. 2. SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs The base of this pyramid is a regular pentagon with of 9 inches and 5 inches and the height of the sides of 4.4 centimeters and an apothem of 3 pyramid is 10 inches. centimeters. The height of the pyramid is 12 centimeters.

ANSWER: 3 ANSWER: 75 in 132 cm3

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the 2. area of the base and h is the height of the pyramid. SOLUTION: The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of The volume of a pyramid is , where B is the the pyramid is 5.2 meters. area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths eSolutions Manual - Powered by Cognero SOLUTION: Page 1

ANSWER: The volume of a pyramid is , where B is the 132 cm3 area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 3. a rectangular pyramid with a height of 5.2 meters meters. The height of the pyramid is 14 meters. and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters. ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth.

ANSWER: 3 5. 62.4 m SOLUTION: 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths The volume of a circular cone is , or SOLUTION: , where B is the area of the base, h is the The volume of a pyramid is , where B is the height of the cone, and r is the radius of the base. area of the base and h is the height of the pyramid. Since the diameter of this cone is 7 inches, the radius The base of this pyramid is a square with sides of 8 is or 3.5 inches. The height of the cone is 4 inches. meters. The height of the pyramid is 14 meters.

ANSWER: ANSWER: 3 3 51.3 in 298.7 m

Find the volume of each cone. Round to the nearest tenth.

5. SOLUTION: SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the Use trigonometry to find the radius r. height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

ANSWER: 3 51.3 in

ANSWER: 168.1 cm3

6. 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the

Use trigonometry to find the radius r. height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: Use the Pythagorean Theorem to find the height h of The volume of a circular cone is , or the cone. Then find its volume.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

So, the height of the cone is 20 meters.

ANSWER: 3 28.1 mm ANSWER: 8. a cone with a slant height of 25 meters and a radius 3 of 15 meters 4712.4 m

SOLUTION: 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth.

Use the Pythagorean Theorem to find the height h of SOLUTION: the cone. Then find its volume. The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

So, the height of the cone is 20 meters.

ANSWER: 3 513,333.3 ft

ANSWER: CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if 4712.4 m3 necessary. 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. 10. SOLUTION: SOLUTION:

The volume of a circular cone is , where B The volume of a pyramid is , where B is the is the area of the base and h is the height of the area of the base and h is the height of the pyramid. cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 605 in3

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. 11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 10.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 105.8 mm

ANSWER: 3 605 in 12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER:

482.1 m3

ANSWER: 3 105.8 mm 13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

12. The base is a hexagon, so we need to make a right tri SOLUTION: determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl The volume of a pyramid is , where B is the 90° triangle.

area of the base and h is the height of the pyramid.

ANSWER: 482.1 m3

The apothem is .

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri ANSWER: determine the apothem. The interior angles of the he 3 120°. The radius bisects the angle, so the right triangl 233.8 cm 90° triangle. 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is . ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters The volume of a pyramid is , where B is the and a right triangle base with a leg 5 centimeters and area of the base and h is the height of the pyramid. hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

ANSWER: 3 35.6 cm

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The length of the other leg of the right triangle is 6 ANSWER: cm. 3 35.6 cm 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. So, the area of the base B is 24 cm2. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the Replace V with 144 and B with 24 in the formula for triangle. the volume of a pyramid and solve for the height h.

Therefore, the height of the triangular pyramid is 18 cm.

ANSWER: 18 cm The length of the other leg of the right triangle is 6 Find the volume of each cone. Round to the cm. nearest tenth.

17. SOLUTION:

2 So, the area of the base B is 24 cm . The volume of a circular cone is ,

Replace V with 144 and B with 24 in the formula for where r is the radius of the base and h is the height the volume of a pyramid and solve for the height h. of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Therefore, the height of the triangular pyramid is 18 cm. 3 ANSWER: Therefore, the volume of the cone is about 235.6 in . 18 cm ANSWER: Find the volume of each cone. Round to the 235.6 in3 nearest tenth.

17. 18. SOLUTION: SOLUTION: The volume of a circular cone is , The volume of a circular cone is , where where r is the radius of the base and h is the height of the cone. r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and

Since the diameter of this cone is 10 inches, the the height is 7.3 centimeters.

radius is or 5 inches. The height of the cone is 9

inches.

Therefore, the volume of the cone is about 134.8 3 cm . 3 Therefore, the volume of the cone is about 235.6 in . ANSWER: 3 ANSWER: 134.8 cm 235.6 in3

19.

18. SOLUTION: SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Use a trigonometric ratio to find the height h of the

cone.

Therefore, the volume of the cone is about 134.8 The volume of a circular cone is , where 3 cm . r is the radius of the base and h is the height of the

ANSWER: cone. The radius of this cone is 8 centimeters.

3 134.8 cm

19. Therefore, the volume of the cone is about 1473.1 3 SOLUTION: cm .

ANSWER: 1473.1 cm3

Use a trigonometric ratio to find the height h of the 20. cone. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. Use trigonometric ratios to find the height h and the radius r of the cone.

Therefore, the volume of the cone is about 1473.1 The volume of a circular cone is , where 3 cm . r is the radius of the base and h is the height of the ANSWER: cone.

3 1473.1 cm

20.

SOLUTION: 3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches Use trigonometric ratios to find the height h and the radius r of the cone. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

3 Therefore, the volume of the cone is about 2.8 ft . 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter ANSWER: SOLUTION: 3 2.8 ft The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean 21. an oblique cone with a diameter of 16 inches and an Theorem to find the height h of the cone. altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

22. a right cone with a slant height of 5.6 centimeters 3 and a radius of 1 centimeter Therefore, the volume of the cone is about 5.8 cm .

SOLUTION: ANSWER: The cone has a radius r of 1 centimeter and a slant 3 height of 5.6 centimeters. Use the Pythagorean 5.8 cm Theorem to find the height h of the cone. 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. 3 Therefore, the volume of the cone is about 5.8 cm . ANSWER: ANSWER: 3 234.6 cm 5.8 cm3 24. CCSS MODELING The Pyramid Arena in 23. SNACKS Approximately how many cubic Memphis, Tennessee, is the third largest pyramid in centimeters of roasted peanuts will completely fill a the world. It is approximately 350 feet tall, and its paper cone that is 14 centimeters high and has a base square base is 600 feet wide. Find the volume of this diameter of 8 centimeters? Round to the nearest pyramid. tenth. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a circular cone is , where area of the base and h is the height of the pyramid. r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular 3 Therefore, the paper cone will hold about 234.6 cm octagonal pyramid with a height of 5 feet. The base of roasted peanuts. has side lengths of 2 feet. What is the volume of the greenhouse? ANSWER: 234.6 cm3

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular Use a trigonometric ratio to find the apothem a. octagonal pyramid with a height of 5 feet. The base

has side lengths of 2 feet. What is the volume of the greenhouse?

The height of this pyramid is 5 feet.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle Therefore, the volume of the greenhouse is about below is 22.5°. 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

Use a trigonometric ratio to find the apothem a.

26. SOLUTION:

Volume of the solid given = Volume of the small The height of this pyramid is 5 feet. cone + Volume of the large cone

Therefore, the volume of the greenhouse is about 32.2 ft3. ANSWER: ANSWER: 471.2 in3 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

27. SOLUTION: 26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 3 3190.6 m

ANSWER: 28. 3 471.2 in SOLUTION:

27.

SOLUTION:

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per ANSWER: cubic foot. What size unit does Sam need to 3 3190.6 m purchase?

28. SOLUTION: SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

The volume of the ceiling is ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units)

required to heat the building. For new construction The total volume is therefore 5000 + 1666.67 = with good insulation, there should be 2 BTUs per 3 cubic foot. What size unit does Sam need to 6666.67 ft . Two BTU's are needed for every cubic purchase? foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay SOLUTION: model of a crystal with a shape that is a composite of The building can be broken down into the rectangular two congruent rectangular pyramids. The base of base and the pyramid ceiling. The volume of the base each pyramid will be 1 by 1.5 inches, and the total is height will be 4 inches.

Determine the volume of the model. Explain why knowing the volume is helpful in this situation.

SOLUTION:

The volume of the ceiling is The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic It tells Marta how much clay is needed to make the foot, so the size of the heating unit Sam should buy is model. 6666.67 × 2 = 13,333 BTUs. ANSWER: 3 ANSWER: 2 in ; It tells Marta how much clay is needed to make the model. 13,333 BTUs 31. CHANGING DIMENSIONS A cone has a radius 30. SCIENCE Marta is studying crystals that grow on of 4 centimeters and a height of 9 centimeters. rock formations.For a project, she is making a clay Describe how each change affects the volume of the model of a crystal with a shape that is a composite of cone. two congruent rectangular pyramids. The base of a. each pyramid will be 1 by 1.5 inches, and the total The height is doubled. height will be 4 inches. b. The radius is doubled. c. Both the radius and the height are doubled. Determine the volume of the model. Explain why SOLUTION: knowing the volume is helpful in this situation. Find the volume of the original cone. Then alter the SOLUTION: values.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

a. Double h.

It tells Marta how much clay is needed to make the model.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model. The volume is doubled. 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. b. Double r. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION:

Find the volume of the original cone. Then alter the 2 values. The volume is multiplied by 2 or 4.

c. Double r and h.

a. Double h. volume is multiplied by 23 or 8.

ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

The volume is doubled. Find each measure. Round to the nearest tenth if necessary. b. Double r. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. 2 Let s be the side length of the base. The volume is multiplied by 2 or 4.

c. Double r and h.

volume is multiplied by 23 or 8.

ANSWER: a. The volume is doubled. 2 The side length of the base is 15 cm. b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8. ANSWER: 15 cm Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic 33. The volume of a cone is 196π cubic inches and the centimeters and a height of 11.5 centimeters. Find height is 12 inches. What is the diameter? the side length of the base. SOLUTION: SOLUTION: The volume of a circular cone is , or The volume of a pyramid is , where B is the , where B is the area of the base, h is the area of the base and h is the height of the pyramid. Let s be the side length of the base. height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The side length of the base is 15 cm. The diameter is 2(7) or 14 inches. ANSWER: ANSWER: 15 cm 14 in. 33. The volume of a cone is 196π cubic inches and the 34. The lateral area of a cone is 71.6 square millimeters height is 12 inches. What is the diameter? and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION: SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The diameter is 2(7) or 14 inches. So, the radius is about 3.8 millimeters. ANSWER: 14 in. Use the Pythagorean Theorem to find the height of the cone. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the The lateral area of a cone is , where r is cone. the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

Therefore, the volume of the cone is about 70.2 3 mm . So, the radius is about 3.8 millimeters. ANSWER: Use the Pythagorean Theorem to find the height of 70.2 mm3 the cone. 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base So, the height of the cone is about 4.64 millimeters. area and/or the height of the pyramid by 5 affects the volume of the pyramid. The volume of a circular cone is , where SOLUTION: a. r is the radius of the base and h is the height of the Use rectangular bases and pick values that

cone. multiply to make 24.

Sample answer:

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. b. The volumes are the same. The volume of a c. ANALYTICAL Explain how multiplying the base pyramid equals one third times the base area times area and/or the height of the pyramid by 5 affects the the height. So, if the base areas of two pyramids are volume of the pyramid. equal and their heights are equal, then their volumes

SOLUTION: are equal. a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

ANSWER: a. Sample answer:

b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and The volumes will only be equal when the radius of the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. the cone is equal to or when . Therefore, the statement is true sometimes if the 36. CCSS ARGUMENTS Determine whether the base area of the cone is 3 times as great as the base following statement is sometimes, always, or never area of the prism. For example, if the base of the true. Justify your reasoning. prism has an area of 10 square units, then its volume The volume of a cone with radius r and height h is 10h cubic units. So, the cone must have a base equals the volume of a prism with height h. area of 30 square units so that its volume is

SOLUTION: or 10h cubic units. The volume of a cone with a radius r and height h is . The volume of a prism with a height of ANSWER: h is where B is the area of the base of the Sometimes; the statement is true if the base area of prism. Set the volumes equal. the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

square units so that its volume is or 10h 38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that cubic units. has the same radius and height as the cone? Explain

37. ERROR ANALYSIS Alexandra and Cornelio are your reasoning. calculating the volume of the cone below. Is either of SOLUTION: them correct? Explain your answer. 3 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

ANSWER: 1704 cm3; The volume of a cylinder is three times as much as the volume of a cone with the same radius and height. SOLUTION: OPEN ENDED The slant height is used for surface area, but the 39. Give an example of a pyramid and height is used for volume. For this cone, the slant a prism that have the same base and the same

height of 13 is provided, and we need to calculate the volume. Explain your reasoning. height before we can calculate the volume. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of Alexandra incorrectly used the slant height. that. So, if a pyramid and prism have the same base, ANSWER: then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of Cornelio; Alexandra incorrectly used the slant height. the prism.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. SOLUTION: 3 1704 cm ; The formula for the volume of a cylinder Set the base areas of the prism and pyramid, and is V= Bh, while the formula for the volume of a cone make the height of the pyramid equal to 3 times the is V = Bh. The volume of a cylinder is three times height of the prism. as much as the volume of a cone with the same radius and height. Sample answer: A square pyramid with a base area of 16 and a ANSWER: height of 12, a prism with a square base area of 16 1704 cm3; The volume of a cylinder is three times as and a height of 4. much as the volume of a cone with the same radius ANSWER: and height. Sample answer: A square pyramid with a base area 39. OPEN ENDED Give an example of a pyramid and of 16 and a height of 12, a prism with a square base a prism that have the same base and the same area of 16 and a height of 4; if a pyramid and prism volume. Explain your reasoning. have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as SOLUTION: great as the height of the prism. The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of 40. WRITING IN MATH Compare and contrast that. So, if a pyramid and prism have the same base, finding volumes of pyramids and cones with finding then in order to have the same volume, the height of volumes of prisms and cylinders. the pyramid must be 3 times as great as the height of SOLUTION: the prism. To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

ANSWER: Set the base areas of the prism and pyramid, and To find the volume of each solid, you must know the make the height of the pyramid equal to 3 times the area of the base and the height. The volume of a height of the prism. pyramid is one third the volume of a prism that has

the same height and base area. The volume of a Sample answer: cone is one third the volume of a cylinder that has the A square pyramid with a base area of 16 and a same height and base area. height of 12, a prism with a square base area of 16 and a height of 4. 41. A conical sand toy has the dimensions as shown ANSWER: below. How many cubic centimeters of sand will it hold when it is filled to the top? Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism. A 12π B 15π 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding C volumes of prisms and cylinders.

SOLUTION: D To find the volume of each solid, you must know the area of the base and the height. The volume of a SOLUTION: pyramid is one third the volume of a prism that has Use the Pythagorean Theorem to find the radius r of the same height and base area. The volume of a the cone. cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

So, the radius of the cone is 3 centimeters.

The volume of a circular cone is , where A 12π r is the radius of the base and h is the height of the B 15π cone.

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone. Therefore, the correct choice is A.

ANSWER: A

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole?

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored Therefore, the correct choice is A. red, blue, orange, and green. The table below shows the results of several spins. What is the experimental ANSWER: probability of the spinner landing on orange? A

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole?

SOLUTION: F

The volume of a pyramid is , where B is the G area of the base and h is the height of the pyramid. H

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

ANSWER: 5.5 ft

ANSWER: F

F 44. SAT/ACT For all A –8 G B x – 4 C H D

J E SOLUTION: SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

So, the correct choice is E.

ANSWER: E Find the volume of each prism.

So, the correct choice is F.

ANSWER: 45. F SOLUTION: The volume of a prism is , where B is the 44. SAT/ACT For all area of the base and h is the height of the prism. The A –8 base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the B x – 4 prism is 6 inches.

C D E

SOLUTION:

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 1008 in3 So, the correct choice is E.

ANSWER: E Find the volume of each prism. 46. SOLUTION: The volume of a prism is , where B is the 45. area of the base and h is the height of the prism. The SOLUTION: base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will The volume of a prism is , where B is the bisect the base. Use the Pythagorean Theorem to area of the base and h is the height of the prism. The find the height of the triangle. base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 3 1008 in So, the height of the triangle is 12 feet. Find the area of the triangle.

46. SOLUTION:

The volume of a prism is , where B is the So, the area of the base B is 60 ft2. area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will The height h of the prism is 19 feet. bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

Therefore, the volume of the prism is 1140 ft3.

ANSWER:

3 1140 ft

So, the height of the triangle is 12 feet. Find the area 47. of the triangle. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 3 426,437.6 m Therefore, the volume of the prism is 1140 ft .

ANSWER: 48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain 3 1140 ft after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. 47. SOLUTION: SOLUTION: To find the entire surface area of the bin, find the The volume of a prism is , where B is the surface area of the conical top and bottom and find area of the base and h is the height of the prism. The the surface area of the cylinder and add them. base of this prism is a rectangle with a length of 79.4 The formula for finding the surface area of a cone is meters and a width of 52.5 meters. The height of the , where is r is the radius and l is the slant height prism is 102.3 meters. of the cone.

Find the slant height of the conical top.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m 48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain

after harvest. The cone at the bottom of the bin

allows the grain to be emptied more easily. Use the Find the slant height of the conical bottom. dimensions in the diagram to find the entire surface The height of the conical bottom is 28 – (5 + 12 + 2)

area of the bin with a conical top and bottom. Write or 9 ft. the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top. The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface

area of the conical bottom.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle The formula for finding the surface area of a cylinder Area of the rectangle = 10(5) is , where is r is the radius and h is the slant = 50 height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface

area of the conical bottom. ANSWER:

2 30.4 cm

ANSWER: 50. SOLUTION: Area of the shaded region = Area of the circle – Find the area of each shaded region. Polygons Area of the hexagon in 50 - 52 are regular.

49. SOLUTION: A regular hexagon has 6 sides, so the measure of the Area of the shaded region = Area of the rectangle – Area of the circle interior angle is . The apothem bisects the Area of the rectangle = 10(5) angle, so we use 30° when using trig to find the = 50 length of the apothem.

ANSWER: 30.4 cm2

50.

SOLUTION:

Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the

interior angle is . The apothem bisects the Now find the area of the hexagon. angle, so we use 30° when using trig to find the length of the apothem.

ANSWER: 2 54.4 in

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

Now find the area of the hexagon.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

ANSWER: The equilateral triangle can be divided into three 2 54.4 in isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. Use a trigonometric ratio to find the value of b. Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet. Now use the area of the isosceles triangles to find Next, find the area of the equilateral triangle. the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3

The equilateral triangle can be divided into three square feet.

isosceles triangles with central angles of or Subtract to find the area of shaded region. 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 Use a trigonometric ratio to find the value of b. 26.6 ft

52. SOLUTION: Now use the area of the isosceles triangles to find The area of the shaded region is the area of the the area of the equilateral triangle. circumscribed circle minus the area of the equilateral

triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the Therefore, the area of the shaded region is about isosceles triangle is 60° and the base is half the 2 length of the side of the equilateral triangle as shown 26.6 ft . below. ANSWER: 2 26.6 ft

52. Use trigonometric ratios to find the values of b and h. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

Divide the equilateral triangle into three isosceles A(shaded region) = A(circumscribed circle) - A triangles with central angles of or 120°. The (equilateral triangle) + A(inscribed circle) angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER:

2 Use trigonometric ratios to find the values of b and 168.2 mm h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

Find the volume of each pyramid.

1. ANSWER: SOLUTION: 132 cm3

The volume of a pyramid is , where B is the 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs SOLUTION: of 9 inches and 5 inches and the height of the pyramid is 10 inches. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 3 75 in ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

2. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 The volume of a pyramid is , where B is the meters. The height of the pyramid is 14 meters. area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth.

ANSWER: 3 132 cm 5. 3. a rectangular pyramid with a height of 5.2 meters SOLUTION: and a base 8 meters by 4.5 meters The volume of a circular cone is , or SOLUTION:

The volume of a pyramid is , where B is the , where B is the area of the base, h is the area of the base and h is the height of the pyramid. height of the cone, and r is the radius of the base. The base of this pyramid is a rectangle with a length Since the diameter of this cone is 7 inches, the radius of 8 meters and a width of 4.5 meters. The height of is or 3.5 inches. The height of the cone is 4 inches. the pyramid is 5.2 meters.

ANSWER: ANSWER: 12-5 Volumes of Pyramids and Cones 3 3 51.3 in 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the 6. area of the base and h is the height of the pyramid. SOLUTION: The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

Use trigonometry to find the radius r.

ANSWER: The volume of a circular cone is , or 298.7 m3 , where B is the area of the base, h is the Find the volume of each cone. Round to the nearest tenth. height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

5. SOLUTION:

The volume of a circular cone is , or ANSWER: 168.1 cm3 , where B is the area of the base, h is the 7. an oblique cone with a height of 10.5 millimeters and height of the cone, and r is the radius of the base. a radius of 1.6 millimeters Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches. SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

ANSWER: 3 51.3 in

eSolutions Manual - Powered by Cognero ANSWER: Page 2 3 6. 28.1 mm SOLUTION: 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

Use trigonometry to find the radius r.

Use the Pythagorean Theorem to find the height h of The volume of a circular cone is , or the cone. Then find its volume.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

So, the height of the cone is 20 meters.

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters ANSWER: SOLUTION: 4712.4 m3 The volume of a circular cone is , or 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical , where B is the area of the base, h is the building. If the height is 100 feet and the area of the height of the cone, and r is the radius of the base. base is about 15,400 square feet, find the volume of The radius of this cone is 1.6 millimeters and the air that the heating and cooling systems would have height is 10.5 millimeters. to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet. ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if Use the Pythagorean Theorem to find the height h of necessary. the cone. Then find its volume.

10. SOLUTION:

So, the height of the cone is 20 meters. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: 3 4712.4 m ANSWER: 3 9. MUSEUMS The sky dome of the National Corvette 605 in Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION: 11. The volume of a circular cone is , where B SOLUTION: is the area of the base and h is the height of the cone. The volume of a pyramid is , where B is the For this cone, the area of the base is 15,400 square area of the base and h is the height of the pyramid. feet and the height is 100 feet.

ANSWER: ANSWER: 3 105.8 mm 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. 12. SOLUTION:

The volume of a pyramid is , where B is the 10. area of the base and h is the height of the pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 482.1 m3

ANSWER: 605 in3

13. SOLUTION:

The volume of a pyramid is , where B is the 11. base and h is the height of the pyramid.

SOLUTION: The base is a hexagon, so we need to make a right tri The volume of a pyramid is , where B is the determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl area of the base and h is the height of the pyramid. 90° triangle.

ANSWER: 3 105.8 mm

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square ANSWER: feet and an altitude of 7 feet 3 482.1 m SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the

base and h is the height of the pyramid. ANSWER:

3 The base is a hexagon, so we need to make a right tri 1376.7 ft determine the apothem. The interior angles of the he 15. a triangular pyramid with a height of 4.8 centimeters 120°. The radius bisects the angle, so the right triangl and a right triangle base with a leg 5 centimeters and 90° triangle. hypotenuse 10.2 centimeters

SOLUTION: Find the height of the right triangle.

The apothem is .

ANSWER: 3 The volume of a pyramid is , where B is the 233.8 cm area of the base and h is the height of the pyramid. 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. ANSWER: SOLUTION: 1376.7 ft3 The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. 15. a triangular pyramid with a height of 4.8 centimeters Use the Pythagorean Theorem to find the other leg a and a right triangle base with a leg 5 centimeters and of the right triangle and then find the area of the hypotenuse 10.2 centimeters triangle. SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

2 The volume of a pyramid is , where B is the So, the area of the base B is 24 cm .

area of the base and h is the height of the pyramid. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

ANSWER: 3 35.6 cm Therefore, the height of the triangular pyramid is 18 16. A triangular pyramid with a right triangle base with a cm. leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. ANSWER: SOLUTION: 18 cm The base of the pyramid is a right triangle with a leg Find the volume of each cone. Round to the of 8 centimeters and a hypotenuse of 10 centimeters. nearest tenth. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 3 So, the area of the base B is 24 cm2. 235.6 in

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and

the height is 7.3 centimeters. Therefore, the height of the triangular pyramid is 18 cm.

ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth. Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 17. 134.8 cm3 SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

19. Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 SOLUTION: inches.

Use a trigonometric ratio to find the height h of the 3 cone. Therefore, the volume of the cone is about 235.6 in . ANSWER: 235.6 in3

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

18.

SOLUTION:

ANSWER: 1473.1 cm3

Therefore, the volume of the cone is about 134.8 3 cm . 20. ANSWER: SOLUTION: 134.8 cm3

19. Use trigonometric ratios to find the height h and the radius r of the cone. SOLUTION:

The volume of a circular cone is , where Use a trigonometric ratio to find the height h of the r is the radius of the base and h is the height of the cone. cone.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. 3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches Therefore, the volume of the cone is about 1473.1 SOLUTION: 3 cm . The volume of a circular cone is , where ANSWER: 3 r is the radius of the base and h is the height of the 1473.1 cm cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

20. SOLUTION:

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 3 Use trigonometric ratios to find the height h and the 1072.3 in radius r of the cone. 22. a right cone with a slant height of 5.6 centimeters

and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER:

2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

3 The volume of a circular cone is , where Therefore, the volume of the cone is about 5.8 cm .

r is the radius of the base and h is the height of the ANSWER: cone. Since the diameter of this cone is 16 inches, 5.8 cm3 the radius is or 8 inches. 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

Therefore, the volume of the cone is about 1072.3 The volume of a circular cone is , where 3 in . r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 ANSWER: 1072.3 in3 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters. 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its

square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: ANSWER: 3 3 5.8 cm 42,000,000 ft

23. SNACKS Approximately how many cubic 25. GARDENING The greenhouse is a regular centimeters of roasted peanuts will completely fill a octagonal pyramid with a height of 5 feet. The base paper cone that is 14 centimeters high and has a base has side lengths of 2 feet. What is the volume of the diameter of 8 centimeters? Round to the nearest greenhouse? tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8

centimeters, the radius is or 4 centimeters. The SOLUTION: height of the cone is 14 centimeters. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°. 3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its Use a trigonometric ratio to find the apothem a. square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the The height of this pyramid is 5 feet. area of the base and h is the height of the pyramid.

ANSWER: 3 42,000,000 ft Therefore, the volume of the greenhouse is about 25. GARDENING The greenhouse is a regular 32.2 ft3. octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the ANSWER: 3 greenhouse? 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

SOLUTION: 26.

The volume of a pyramid is , where B is the SOLUTION: Volume of the solid given = Volume of the small area of the base and h is the height of the pyramid. cone + Volume of the large cone The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Use a trigonometric ratio to find the apothem a. ANSWER:

471.2 in3

The height of this pyramid is 5 feet.

27. SOLUTION:

Therefore, the volume of the greenhouse is about 32.2 ft3. ANSWER: ANSWER: 3 3 3190.6 m 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

28. SOLUTION: 26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction ANSWER: with good insulation, there should be 2 BTUs per 3 cubic foot. What size unit does Sam need to 471.2 in purchase?

27. SOLUTION: SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

ANSWER: The volume of the ceiling is 3 3190.6 m

The total volume is therefore 5000 + 1666.67 = 3 28. 6666.67 ft . Two BTU's are needed for every cubic SOLUTION: foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

3 height will be 4 inches. 7698.5 cm Determine the volume of the model. Explain why 29. HEATING Sam is building an art studio in her knowing the volume is helpful in this situation. backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) SOLUTION: required to heat the building. For new construction with good insulation, there should be 2 BTUs per The volume of a pyramid is , where B is the cubic foot. What size unit does Sam need to

purchase? area of the base and h is the height of the pyramid.

It tells Marta how much clay is needed to make the SOLUTION: model. The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base ANSWER: is 3 2 in ; It tells Marta how much clay is needed to

make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. The volume of the ceiling is a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs a. Double h. 30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches.

Determine the volume of the model. Explain why The volume is doubled. knowing the volume is helpful in this situation. b. SOLUTION: Double r.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

2 The volume is multiplied by 2 or 4.

c. Double r and h. It tells Marta how much clay is needed to make the model.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius volume is multiplied by 23 or 8. of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the ANSWER: cone. a. The volume is doubled. a. The height is doubled. 2 b. The volume is multiplied by 2 or 4. b. The radius is doubled. c. 3 c. Both the radius and the height are doubled. The volume is multiplied by 2 or 8. SOLUTION: Find each measure. Round to the nearest tenth Find the volume of the original cone. Then alter the if necessary. values. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

a. Double h.

The volume is doubled.

b. Double r.

The side length of the base is 15 cm.

ANSWER: 15 cm

2 The volume is multiplied by 2 or 4. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? c. Double r and h. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4

3 centimeters. volume is multiplied by 2 or 8. ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base.

SOLUTION: The diameter is 2(7) or 14 inches.

The volume of a pyramid is , where B is the ANSWER: area of the base and h is the height of the pyramid. 14 in. Let s be the side length of the base. 34. The lateral area of a cone is 71.6 square millimeters

and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The lateral area of a cone is , where r is The side length of the base is 15 cm. the radius and is the slant height of the cone. ANSWER: Replace L with 71.6 and with 6, then solve for the radius r. 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or So, the radius is about 3.8 millimeters.

, where B is the area of the base, h is the Use the Pythagorean Theorem to find the height of height of the cone, and r is the radius of the base. the cone. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The diameter is 2(7) or 14 inches.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? Therefore, the volume of the cone is about 70.2 SOLUTION: 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. The lateral area of a cone is , where r is b. VERBAL What is true about the volumes of the the radius and is the slant height of the cone. two pyramids that you drew? Explain. Replace L with 71.6 and with 6, then solve for the c. ANALYTICAL Explain how multiplying the base radius r. area and/or the height of the pyramid by 5 affects the

volume of the pyramid.

SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer: So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

b. The volumes are the same. The volume of a

pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal.

Therefore, the volume of the cone is about 70.2 3 mm . ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the c. If the base area is multiplied by 5, the volume is two pyramids that you drew? Explain. multiplied by 5. If the height is multiplied by 5, the c. ANALYTICAL Explain how multiplying the base volume is multiplied by 5. If both the base area and area and/or the height of the pyramid by 5 affects the the height are multiplied by 5, the volume is multiplied

volume of the pyramid. by 5 · 5 or 25. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

ANSWER: a. Sample answer:

or 10h cubic units.

ANSWER: b. The volumes are the same. The volume of a Sometimes; the statement is true if the base area of pyramid equals one third times the base area times the cone is 3 times as great as the base area of the the height. So, if the base areas of two pyramids are prism. For example, if the base of the prism has an equal and their heights are equal, then their volumes area of 10 square units, then its volume is 10h cubic are equal. units. So, the cone must have a base area of 30 c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the square units so that its volume is or 10h volume is multiplied by 5. If both the base area and cubic units. the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. 37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of 36. CCSS ARGUMENTS Determine whether the them correct? Explain your answer. following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h.

SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal. SOLUTION: The slant height is used for surface area, but the height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic The volumes will only be equal when the radius of centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain the cone is equal to or when . your reasoning. Therefore, the statement is true sometimes if the SOLUTION: base area of the cone is 3 times as great as the base 3 area of the prism. For example, if the base of the 1704 cm ; The formula for the volume of a cylinder prism has an area of 10 square units, then its volume is V= Bh, while the formula for the volume of a cone is 10h cubic units. So, the cone must have a base is V = Bh. The volume of a cylinder is three times area of 30 square units so that its volume is as much as the volume of a cone with the same

or 10h cubic units. radius and height. ANSWER: ANSWER: 1704 cm3; The volume of a cylinder is three times as Sometimes; the statement is true if the base area of much as the volume of a cone with the same radius the cone is 3 times as great as the base area of the and height. prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic 39. OPEN ENDED Give an example of a pyramid and units. So, the cone must have a base area of 30 a prism that have the same base and the same

square units so that its volume is or 10h volume. Explain your reasoning. cubic units. SOLUTION: The formula for volume of a prism is V = Bh and the 37. ERROR ANALYSIS Alexandra and Cornelio are formula for the volume of a pyramid is one-third of calculating the volume of the cone below. Is either of that. So, if a pyramid and prism have the same base, them correct? Explain your answer. then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the SOLUTION: height of the prism. The slant height is used for surface area, but the height is used for volume. For this cone, the slant Sample answer: height of 13 is provided, and we need to calculate the A square pyramid with a base area of 16 and a height before we can calculate the volume. height of 12, a prism with a square base area of 16 and a height of 4.

Alexandra incorrectly used the slant height. ANSWER: Sample answer: A square pyramid with a base area ANSWER: of 16 and a height of 12, a prism with a square base Cornelio; Alexandra incorrectly used the slant height. area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same 38. REASONING A cone has a volume of 568 cubic volume, the height of the pyramid must be 3 times as centimeters. What is the volume of a cylinder that great as the height of the prism. has the same radius and height as the cone? Explain your reasoning. 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding SOLUTION: volumes of prisms and cylinders. 3 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone SOLUTION: To find the volume of each solid, you must know the is V = Bh. The volume of a cylinder is three times area of the base and the height. The volume of a as much as the volume of a cone with the same pyramid is one third the volume of a prism that has radius and height. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the ANSWER: same height and base area. 1704 cm3; The volume of a cylinder is three times as much as the volume of a cone with the same radius ANSWER: and height. To find the volume of each solid, you must know the area of the base and the height. The volume of a 39. OPEN ENDED Give an example of a pyramid and pyramid is one third the volume of a prism that has a prism that have the same base and the same the same height and base area. The volume of a volume. Explain your reasoning. cone is one third the volume of a cylinder that has the same height and base area. SOLUTION: The formula for volume of a prism is V = Bh and the 41. A conical sand toy has the dimensions as shown formula for the volume of a pyramid is one-third of below. How many cubic centimeters of sand will it that. So, if a pyramid and prism have the same base, hold when it is filled to the top? then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

A 12π B 15π

Set the base areas of the prism and pyramid, and D make the height of the pyramid equal to 3 times the height of the prism. SOLUTION: Use the Pythagorean Theorem to find the radius r of Sample answer: the cone. A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

ANSWER: Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders.

SOLUTION: So, the radius of the cone is 3 centimeters. To find the volume of each solid, you must know the area of the base and the height. The volume of a The volume of a circular cone is , where pyramid is one third the volume of a prism that has r is the radius of the base and h is the height of the the same height and base area. The volume of a cone. cone is one third the volume of a cylinder that has the same height and base area.

ANSWER: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the Therefore, the correct choice is A. same height and base area. ANSWER: 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it A hold when it is filled to the top? 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION: A 12π B 15π The volume of a pyramid is , where B is the

C area of the base and h is the height of the pyramid.

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the radius of the cone is 3 centimeters.

The volume of a circular cone is , where F r is the radius of the base and h is the height of the cone. G

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 Therefore, the correct choice is A. green} Number of possible outcomes : 25 ANSWER: Favorable outcomes: {5 orange} A Number of favorable outcomes: 5

SHORT RESPONSE 42. Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION:

The volume of a pyramid is , where B is the So, the correct choice is F. area of the base and h is the height of the pyramid. ANSWER: F

44. SAT/ACT For all

A –8 B x – 4 C D ANSWER: E 5.5 ft SOLUTION: 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the correct choice is E.

ANSWER: E

F Find the volume of each prism.

H 45.

J SOLUTION: The volume of a prism is , where B is the SOLUTION: area of the base and h is the height of the prism. The Possible outcomes: {6 red, 4 blue, 5 orange, 10 base of this prism is a rectangle with a length of 14 green} inches and a width of 12 inches. The height h of the Number of possible outcomes : 25 prism is 6 inches. Favorable outcomes: {5 orange} Number of favorable outcomes: 5

3 Therefore, the volume of the prism is 1008 in .

ANSWER: So, the correct choice is F. 1008 in3 ANSWER: F

44. SAT/ACT For all

A –8 46. B x – 4 SOLUTION: C The volume of a prism is , where B is the D area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base E of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to SOLUTION: find the height of the triangle.

So, the correct choice is E.

ANSWER: E Find the volume of each prism.

So, the height of the triangle is 12 feet. Find the area 45. of the triangle.

prism is 6 inches.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1008 in .

ANSWER: Therefore, the volume of the prism is 1140 ft3. 1008 in3 ANSWER: 3 1140 ft

46. SOLUTION: The volume of a prism is , where B is the 47. area of the base and h is the height of the prism. The SOLUTION: base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will The volume of a prism is , where B is the bisect the base. Use the Pythagorean Theorem to area of the base and h is the height of the prism. The find the height of the triangle. base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6

m3.

ANSWER: 3 426,437.6 m

So, the height of the triangle is 12 feet. Find the area 48. FARMING The picture shows a combination of the triangle. hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. 2 So, the area of the base B is 60 ft . SOLUTION: To find the entire surface area of the bin, find the The height h of the prism is 19 feet. surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

3 Find the slant height of the conical top. Therefore, the volume of the prism is 1140 ft . ANSWER: 3 1140 ft

47. SOLUTION: The volume of a prism is , where B is the Find the slant height of the conical bottom. area of the base and h is the height of the prism. The The height of the conical bottom is 28 – (5 + 12 + 2) base of this prism is a rectangle with a length of 79.4 or 9 ft. meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant 48. FARMING The picture shows a combination height of the cylinder. hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface Surface area of the bin = Surface area of the area of the bin with a conical top and bottom. Write cylinder + Surface area of the conical top + surface the exact answer and the answer rounded to the area of the conical bottom. nearest square foot. Refer to the photo on Page 863.

SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is ANSWER: , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top. Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

ANSWER: 30.4 cm2

50. SOLUTION: Area of the shaded region = Area of the circle –

The formula for finding the surface area of a cylinder Area of the hexagon is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the A regular hexagon has 6 sides, so the measure of the cylinder + Surface area of the conical top + surface interior angle is . The apothem bisects the

area of the conical bottom. angle, so we use 30° when using trig to find the length of the apothem.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Now find the area of the hexagon. ANSWER: 30.4 cm2

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon ANSWER: 2 54.4 in

A regular hexagon has 6 sides, so the measure of the 51. interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the SOLUTION: length of the apothem. The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find ANSWER: the area of the equilateral triangle. 2 54.4 in

51. So, the area of the equilateral triangle is about 67.3 SOLUTION: square feet. The shaded area is the area of the equilateral triangle less the area of the inscribed circle. Subtract to find the area of shaded region.

Find the area of the circle with a radius of 3.6 feet.

Therefore, the area of the shaded region is about 2 26.6 ft . So, the area of the circle is about 40.7 square feet. ANSWER: 2 Next, find the area of the equilateral triangle. 26.6 ft

52. SOLUTION: The area of the shaded region is the area of the The equilateral triangle can be divided into three circumscribed circle minus the area of the equilateral isosceles triangles with central angles of or triangle plus the area of the inscribed circle. Let b 120°, so the angle in the triangle created by the represent the length of each side of the equilateral height of the isosceles triangle is 60° and the base is triangle and h represent the radius of the inscribed half the length b of the side of the equilateral triangle. circle.

Use a trigonometric ratio to find the value of b. Divide the equilateral triangle into three isosceles

triangles with central angles of or 120°. The

angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown

below. Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and h. So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 2 The area of the equilateral triangle will equal three 26.6 ft . times the area of any of the isosceles triangles. Find ANSWER: the area of the shaded region. 2 26.6 ft A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

52.

SOLUTION: The area of the shaded region is the area of the Therefore, the area of the shaded region is about circumscribed circle minus the area of the equilateral 168.2 mm2. triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed ANSWER: circle. 2 168.2 mm

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

Find the volume of each pyramid.

ANSWER: 3 75 in 1. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs 2. of 9 inches and 5 inches and the height of the pyramid is 10 inches. SOLUTION:

ANSWER: 3 75 in

ANSWER: 132 cm3 2. SOLUTION: 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with The volume of a pyramid is , where B is the sides of 4.4 centimeters and an apothem of 3 area of the base and h is the height of the pyramid. centimeters. The height of the pyramid is 12 The base of this pyramid is a rectangle with a length centimeters. of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 3 ANSWER: 62.4 m 3 132 cm 4. a square pyramid with a height of 14 meters and a

3. a rectangular pyramid with a height of 5.2 meters base with 8-meter side lengths and a base 8 meters by 4.5 meters SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8

The base of this pyramid is a rectangle with a length meters. The height of the pyramid is 14 meters. of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 298.7 m3 ANSWER: 3 Find the volume of each cone. Round to the 62.4 m nearest tenth. 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the 5. area of the base and h is the height of the pyramid. SOLUTION: The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth. ANSWER: 3 51.3 in

5. SOLUTION:

The volume of a circular cone is , or 6. , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

Use trigonometry to find the radius r.

The volume of a circular cone is , or ANSWER: 3 , where B is the area of the base, h is the 51.3 in height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

6. SOLUTION: ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters Use trigonometry to find the radius r. SOLUTION: The volume of a circular cone is , or

The volume of a circular cone is , or , where B is the area of the base, h is the height of the cone, and r is the radius of the base. , where B is the area of the base, h is the The radius of this cone is 1.6 millimeters and the height of the cone, and r is the radius of the base. height is 10.5 millimeters. The height of the cone is 11.5 centimeters.

ANSWER: ANSWER: 12-5 Volumes of Pyramids and Cones 3 168.1 cm3 28.1 mm

7. an oblique cone with a height of 10.5 millimeters and 8. a cone with a slant height of 25 meters and a radius a radius of 1.6 millimeters of 15 meters SOLUTION: SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base.

The radius of this cone is 1.6 millimeters and the Use the Pythagorean Theorem to find the height h of height is 10.5 millimeters. the cone. Then find its volume.

ANSWER: So, the height of the cone is 20 meters. 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

ANSWER: 4712.4 m3

MUSEUMS Use the Pythagorean Theorem to find the height h of 9. The sky dome of the National Corvette the cone. Then find its volume. Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B

So, the height of the cone is 20 meters. is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 3 eSolutions4712.4Manual m - Powered by Cognero Page 3 9. MUSEUMS The sky dome of the National Corvette ANSWER: 3 Museum in Bowling Green, Kentucky, is a conical 513,333.3 ft building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of CCSS SENSE-MAKING Find the volume of air that the heating and cooling systems would have each pyramid. Round to the nearest tenth if to accommodate. Round to the nearest tenth. necessary. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. 10. For this cone, the area of the base is 15,400 square feet and the height is 100 feet. SOLUTION: The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: 3 513,333.3 ft ANSWER: CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if 605 in3 necessary.

10. 11. SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 3 3 605 in 105.8 mm

12. 11. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 482.1 m3 3 105.8 mm

12. 13. SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 482.1 m3

13. SOLUTION: The apothem is . The volume of a pyramid is , where B is the

base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters ANSWER: and a right triangle base with a leg 5 centimeters and 3 hypotenuse 10.2 centimeters 233.8 cm SOLUTION: 14. a pentagonal pyramid with a base area of 590 square Find the height of the right triangle. feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the The volume of a pyramid is , where B is the triangle. area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. The length of the other leg of the right triangle is 6 SOLUTION: cm. The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The length of the other leg of the right triangle is 6 cm. Therefore, the height of the triangular pyramid is 18 cm. ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. 17.

SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 Therefore, the height of the triangular pyramid is 18 inches. cm.

ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 17. 235.6 in3 SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. 18. Since the diameter of this cone is 10 inches, the SOLUTION: radius is or 5 inches. The height of the cone is 9 inches. The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3 Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 134.8 cm3

18. SOLUTION:

The volume of a circular cone is , where 19. r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and SOLUTION: the height is 7.3 centimeters.

Use a trigonometric ratio to find the height h of the Therefore, the volume of the cone is about 134.8 cone. 3 cm .

ANSWER: 134.8 cm3

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

19. SOLUTION:

Therefore, the volume of the cone is about 1473.1 3 cm .

ANSWER: Use a trigonometric ratio to find the height h of the 3 cone. 1473.1 cm

20. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Use trigonometric ratios to find the height h and the radius r of the cone.

Therefore, the volume of the cone is about 1473.1 3 cm .

ANSWER: 1473.1 cm3

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. 20.

SOLUTION:

3 Therefore, the volume of the cone is about 2.8 ft . Use trigonometric ratios to find the height h and the radius r of the cone. ANSWER:

2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where The volume of a circular cone is , where r is the radius of the base and h is the height of the r is the radius of the base and h is the height of the cone. cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

3 Therefore, the volume of the cone is about 2.8 ft . Therefore, the volume of the cone is about 1072.3 3 in . ANSWER: 2.8 ft3 ANSWER: 1072.3 in3 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: SOLUTION: The volume of a circular cone is , where The cone has a radius r of 1 centimeter and a slant r is the radius of the base and h is the height of the height of 5.6 centimeters. Use the Pythagorean cone. Since the diameter of this cone is 16 inches, Theorem to find the height h of the cone. the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

22. a right cone with a slant height of 5.6 centimeters

and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. 3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3 3 Therefore, the paper cone will hold about 234.6 cm 23. SNACKS Approximately how many cubic of roasted peanuts. centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base ANSWER: diameter of 8 centimeters? Round to the nearest 234.6 cm3 tenth. SOLUTION: 24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in The volume of a circular cone is , where the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this r is the radius of the base and h is the height of the pyramid. cone. Since the diameter of the cone is 8 SOLUTION: centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. ANSWER: 3 ANSWER: 42,000,000 ft 234.6 cm3 25. GARDENING The greenhouse is a regular 24. CCSS MODELING The Pyramid Arena in octagonal pyramid with a height of 5 feet. The base Memphis, Tennessee, is the third largest pyramid in has side lengths of 2 feet. What is the volume of the the world. It is approximately 350 feet tall, and its greenhouse? square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is ANSWER: or 45°, so the angle formed in the triangle 3 below is 22.5°. 42,000,000 ft 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Use a trigonometric ratio to find the apothem a.

SOLUTION:

The volume of a pyramid is , where B is the The height of this pyramid is 5 feet.

area of the base and h is the height of the pyramid.

The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Therefore, the volume of the greenhouse is about 32.2 ft3. ANSWER: Use a trigonometric ratio to find the apothem a. 3 32.2 ft Find the volume of each solid. Round to the nearest tenth.

The height of this pyramid is 5 feet.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth. ANSWER: 471.2 in3

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

27. SOLUTION:

ANSWER: 3 ANSWER: 3190.6 m 471.2 in3

28. SOLUTION: 27. SOLUTION:

ANSWER: ANSWER: 7698.5 cm3 3 3190.6 m 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase? 28. SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) The volume of the ceiling is required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. SOLUTION: The building can be broken down into the rectangular ANSWER: base and the pyramid ceiling. The volume of the base is 13,333 BTUs

each pyramid will be 1 by 1.5 inches, and the total The volume of the ceiling is height will be 4 inches.

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the

The total volume is therefore 5000 + 1666.67 = area of the base and h is the height of the pyramid. 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs It tells Marta how much clay is needed to make the 30. SCIENCE Marta is studying crystals that grow on model. rock formations.For a project, she is making a clay ANSWER: model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of 2 in3; It tells Marta how much clay is needed to each pyramid will be 1 by 1.5 inches, and the total make the model. height will be 4 inches. 31. CHANGING DIMENSIONS A cone has a radius Determine the volume of the model. Explain why of 4 centimeters and a height of 9 centimeters. knowing the volume is helpful in this situation. Describe how each change affects the volume of the cone. SOLUTION: a. The height is doubled. b. The volume of a pyramid is , where B is the The radius is doubled. c. Both the radius and the height are doubled. area of the base and h is the height of the pyramid. SOLUTION: Find the volume of the original cone. Then alter the values.

It tells Marta how much clay is needed to make the model.

ANSWER: 2 in3; It tells Marta how much clay is needed to a. Double h. make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. The volume is doubled. c. Both the radius and the height are doubled. b. SOLUTION: Double r. Find the volume of the original cone. Then alter the values.

2 The volume is multiplied by 2 or 4.

c. Double r and h. a. Double h.

volume is multiplied by 23 or 8. The volume is doubled. ANSWER: b. Double r. a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find 2 The volume is multiplied by 2 or 4. the side length of the base. SOLUTION: c. Double r and h. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

volume is multiplied by 23 or 8.

ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find The side length of the base is 15 cm. the side length of the base. ANSWER: SOLUTION: 15 cm The volume of a pyramid is , where B is the 33. The volume of a cone is 196π cubic inches and the area of the base and h is the height of the pyramid. height is 12 inches. What is the diameter? Let s be the side length of the base. SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? The diameter is 2(7) or 14 inches. SOLUTION: ANSWER: The volume of a circular cone is , or 14 in.

, where B is the area of the base, h is the 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the height of the cone, and r is the radius of the base. volume of the cone? Since the diameter is 8 centimeters, the radius is 4 centimeters. SOLUTION:

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r. The diameter is 2(7) or 14 inches. ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the

volume of the cone? So, the radius is about 3.8 millimeters. SOLUTION: Use the Pythagorean Theorem to find the height of the cone.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. So, the height of the cone is about 4.64 millimeters. Replace L with 71.6 and with 6, then solve for the radius r. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone. Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. So, the height of the cone is about 4.64 millimeters. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. The volume of a circular cone is , where b. VERBAL What is true about the volumes of the r is the radius of the base and h is the height of the two pyramids that you drew? Explain. cone. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer: Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24. b. The volumes are the same. The volume of a pyramid equals one third times the base area times Sample answer: the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal.

ANSWER: a. Sample answer:

b. The volumes are the same. The volume of a ANSWER: pyramid equals one third times the base area times a. Sample answer: the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal. b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h The volumes will only be equal when the radius of equals the volume of a prism with height h. the cone is equal to or when . SOLUTION: Therefore, the statement is true sometimes if the The volume of a cone with a radius r and height h is base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the . The volume of a prism with a height of prism has an area of 10 square units, then its volume h is where B is the area of the base of the is 10h cubic units. So, the cone must have a base prism. Set the volumes equal. area of 30 square units so that its volume is

or 10h cubic units.

The volumes will only be equal when the radius of 37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of the cone is equal to or when . them correct? Explain your answer. Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

ANSWER: Sometimes; the statement is true if the base area of SOLUTION: the cone is 3 times as great as the base area of the The slant height is used for surface area, but the prism. For example, if the base of the prism has an height is used for volume. For this cone, the slant area of 10 square units, then its volume is 10h cubic height of 13 is provided, and we need to calculate the units. So, the cone must have a base area of 30 height before we can calculate the volume. square units so that its volume is or 10h

cubic units. Alexandra incorrectly used the slant height.

37. ERROR ANALYSIS Alexandra and Cornelio are ANSWER: calculating the volume of the cone below. Is either of Cornelio; Alexandra incorrectly used the slant height. them correct? Explain your answer. 38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. SOLUTION: 3 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times

as much as the volume of a cone with the same SOLUTION: radius and height. The slant height is used for surface area, but the height is used for volume. For this cone, the slant ANSWER: height of 13 is provided, and we need to calculate the 1704 cm3; The volume of a cylinder is three times as height before we can calculate the volume. much as the volume of a cone with the same radius and height.

Alexandra incorrectly used the slant height. 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same ANSWER: volume. Explain your reasoning. Cornelio; Alexandra incorrectly used the slant height. SOLUTION: 38. REASONING A cone has a volume of 568 cubic The formula for volume of a prism is V = Bh and the centimeters. What is the volume of a cylinder that formula for the volume of a pyramid is one-third of has the same radius and height as the cone? Explain that. So, if a pyramid and prism have the same base, your reasoning. then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of SOLUTION: the prism. 3 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

ANSWER: 3 Set the base areas of the prism and pyramid, and 1704 cm ; The volume of a cylinder is three times as make the height of the pyramid equal to 3 times the much as the volume of a cone with the same radius height of the prism. and height. Sample answer: 39. OPEN ENDED Give an example of a pyramid and A square pyramid with a base area of 16 and a a prism that have the same base and the same height of 12, a prism with a square base area of 16 volume. Explain your reasoning. and a height of 4.

SOLUTION: ANSWER: The formula for volume of a prism is V = Bh and the Sample answer: A square pyramid with a base area formula for the volume of a pyramid is one-third of of 16 and a height of 12, a prism with a square base that. So, if a pyramid and prism have the same base, area of 16 and a height of 4; if a pyramid and prism then in order to have the same volume, the height of have the same base, then in order to have the same the pyramid must be 3 times as great as the height of volume, the height of the pyramid must be 3 times as

the prism. great as the height of the prism.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the Set the base areas of the prism and pyramid, and area of the base and the height. The volume of a make the height of the pyramid equal to 3 times the pyramid is one third the volume of a prism that has height of the prism. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the Sample answer: same height and base area. A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 ANSWER: and a height of 4. To find the volume of each solid, you must know the area of the base and the height. The volume of a ANSWER: pyramid is one third the volume of a prism that has Sample answer: A square pyramid with a base area the same height and base area. The volume of a of 16 and a height of 12, a prism with a square base cone is one third the volume of a cylinder that has the area of 16 and a height of 4; if a pyramid and prism same height and base area. have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as 41. A conical sand toy has the dimensions as shown great as the height of the prism. below. How many cubic centimeters of sand will it hold when it is filled to the top? 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the A 12π area of the base and the height. The volume of a B 15π pyramid is one third the volume of a prism that has the same height and base area. The volume of a C cone is one third the volume of a cylinder that has the same height and base area. D

ANSWER: SOLUTION: To find the volume of each solid, you must know the Use the Pythagorean Theorem to find the radius r of area of the base and the height. The volume of a the cone. pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

A 12π B 15π

C So, the radius of the cone is 3 centimeters.

D The volume of a circular cone is , where r is the radius of the base and h is the height of the SOLUTION: cone. Use the Pythagorean Theorem to find the radius r of the cone.

Therefore, the correct choice is A.

ANSWER:

A 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole?

So, the radius of the cone is 3 centimeters. SOLUTION:

The volume of a circular cone is , where The volume of a pyramid is , where B is the r is the radius of the base and h is the height of the area of the base and h is the height of the pyramid. cone.

Therefore, the correct choice is A.

ANSWER: A ANSWER: 5.5 ft 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 43. PROBABILITY A spinner has sections colored 6 feet by 8 feet. If the tent holds 88 cubic feet of air, red, blue, orange, and green. The table below shows how tall is the tent’s center pole? the results of several spins. What is the experimental probability of the spinner landing on orange? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: SOLUTION: 5.5 ft Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} 43. PROBABILITY A spinner has sections colored Number of possible outcomes : 25 red, blue, orange, and green. The table below shows Favorable outcomes: {5 orange} the results of several spins. What is the experimental Number of favorable outcomes: 5 probability of the spinner landing on orange?

F So, the correct choice is F.

G ANSWER: F H

44. SAT/ACT For all J A –8 SOLUTION: B x – 4 Possible outcomes: {6 red, 4 blue, 5 orange, 10 C green} Number of possible outcomes : 25 D Favorable outcomes: {5 orange} Number of favorable outcomes: 5 E

SOLUTION:

So, the correct choice is E. So, the correct choice is F. ANSWER: ANSWER: E F Find the volume of each prism.

44. SAT/ACT For all

A –8 B x – 4 45. C SOLUTION: D The volume of a prism is , where B is the area of the base and h is the height of the prism. The

E base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the SOLUTION: prism is 6 inches.

So, the correct choice is E.

ANSWER: 3 E Therefore, the volume of the prism is 1008 in . Find the volume of each prism. ANSWER: 1008 in3

45. SOLUTION: The volume of a prism is , where B is the 46. area of the base and h is the height of the prism. The SOLUTION: base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the The volume of a prism is , where B is the prism is 6 inches. area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

3 Therefore, the volume of the prism is 1008 in .

ANSWER:

1008 in3

46. So, the height of the triangle is 12 feet. Find the area SOLUTION: of the triangle.

The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 So, the height of the triangle is 12 feet. Find the area 1140 ft of the triangle.

47.

SOLUTION: So, the area of the base B is 60 ft2. The volume of a prism is , where B is the area of the base and h is the height of the prism. The The height h of the prism is 19 feet. base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the

prism is 102.3 meters.

3 Therefore, the volume of the prism is 1140 ft . ANSWER: Therefore, the volume of the prism is about 426,437.6 3 3 m . 1140 ft ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain 47. after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the SOLUTION: dimensions in the diagram to find the entire surface The volume of a prism is , where B is the area of the bin with a conical top and bottom. Write area of the base and h is the height of the prism. The the exact answer and the answer rounded to the base of this prism is a rectangle with a length of 79.4 nearest square foot. meters and a width of 52.5 meters. The height of the Refer to the photo on Page 863. prism is 102.3 meters. SOLUTION:

To find the entire surface area of the bin, find the

surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height Therefore, the volume of the prism is about 426,437.6 of the cone. m3. Find the slant height of the conical top. ANSWER: 3 426,437.6 m

the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant

height of the cylinder.

Find the slant height of the conical bottom. Surface area of the bin = Surface area of the The height of the conical bottom is 28 – (5 + 12 + 2) cylinder + Surface area of the conical top + surface or 9 ft. area of the conical bottom.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder. 49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Surface area of the bin = Surface area of the Area of the rectangle = 10(5) cylinder + Surface area of the conical top + surface = 50 area of the conical bottom.

ANSWER: ANSWER: 30.4 cm2

Find the area of each shaded region. Polygons in 50 - 52 are regular.

50.

49. SOLUTION: Area of the shaded region = Area of the circle – SOLUTION: Area of the hexagon Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

A regular hexagon has 6 sides, so the measure of the

interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem. ANSWER: 30.4 cm2

50.

SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Now find the area of the hexagon.

ANSWER: 2 54.4 in

51. SOLUTION: Now find the area of the hexagon. The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

ANSWER: 2 54.4 in

51. The equilateral triangle can be divided into three SOLUTION: isosceles triangles with central angles of or The shaded area is the area of the equilateral triangle 120°, so the angle in the triangle created by the less the area of the inscribed circle. height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet. Use a trigonometric ratio to find the value of b.

Next, find the area of the equilateral triangle.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Subtract to find the area of shaded region.

Use a trigonometric ratio to find the value of b.

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft Now use the area of the isosceles triangles to find the area of the equilateral triangle.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral So, the area of the equilateral triangle is about 67.3 triangle plus the area of the inscribed circle. Let b square feet. represent the length of each side of the equilateral

triangle and h represent the radius of the inscribed

Subtract to find the area of shaded region. circle.

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: Divide the equilateral triangle into three isosceles 2 triangles with central angles of or 120°. The 26.6 ft angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral

triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral Use trigonometric ratios to find the values of b and triangle and h represent the radius of the inscribed h. circle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the The area of the equilateral triangle will equal three isosceles triangle is 60° and the base is half the times the area of any of the isosceles triangles. Find length of the side of the equilateral triangle as shown the area of the shaded region. below. A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Use trigonometric ratios to find the values of b and h. Therefore, the area of the shaded region is about 2 168.2 mm .

ANSWER: 2 168.2 mm

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

Find the volume of each pyramid.

ANSWER: 132 cm3 1. 3. a rectangular pyramid with a height of 5.2 meters SOLUTION: and a base 8 meters by 4.5 meters The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the area of the base and h is the height of the pyramid. pyramid is 10 inches. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: ANSWER: 3 3 62.4 m 75 in 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

2. The base of this pyramid is a square with sides of 8 SOLUTION: meters. The height of the pyramid is 14 meters.

298.7 m3

Find the volume of each cone. Round to the nearest tenth.

5. ANSWER: 132 cm3 SOLUTION:

3. a rectangular pyramid with a height of 5.2 meters The volume of a circular cone is , or and a base 8 meters by 4.5 meters , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. The volume of a pyramid is , where B is the Since the diameter of this cone is 7 inches, the radius area of the base and h is the height of the pyramid. is or 3.5 inches. The height of the cone is 4 inches. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 3 51.3 in ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths

SOLUTION: 6. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

Use trigonometry to find the radius r.

The volume of a circular cone is , or

ANSWER: , where B is the area of the base, h is the 298.7 m3 height of the cone, and r is the radius of the base. Find the volume of each cone. Round to the The height of the cone is 11.5 centimeters. nearest tenth.

5. ANSWER: SOLUTION: 168.1 cm3 The volume of a circular cone is , or 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius The volume of a circular cone is , or is or 3.5 inches. The height of the cone is 4 inches. , where B is the area of the base, h is the

height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

ANSWER: 3 51.3 in

ANSWER: 3 28.1 mm

6. 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: SOLUTION:

Use trigonometry to find the radius r. Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

So, the height of the cone is 20 meters.

ANSWER: 168.1 cm3 ANSWER: 7. an oblique cone with a height of 10.5 millimeters and 3 a radius of 1.6 millimeters 4712.4 m SOLUTION: 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical The volume of a circular cone is , or building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of , where B is the area of the base, h is the air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the SOLUTION: height is 10.5 millimeters. The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

10. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. So, the height of the cone is 20 meters.

ANSWER: ANSWER: 12-5 Volumes of Pyramids and Cones 605 in3 4712.4 m3

ANSWER: 3 105.8 mm

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if 12. necessary. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER:

482.1 m3

ANSWER: 605 in3 13. SOLUTION: eSolutions Manual - Powered by Cognero The volume of a pyramid is , where B isPage the4 base and h is the height of the pyramid.

11. The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he SOLUTION: 120°. The radius bisects the angle, so the right triangl 90° triangle. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 105.8 mm

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION: ANSWER: 482.1 m3 The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the ANSWER: 1376.7 ft3 base and h is the height of the pyramid. 15. a triangular pyramid with a height of 4.8 centimeters The base is a hexagon, so we need to make a right tri and a right triangle base with a leg 5 centimeters and determine the apothem. The interior angles of the he hypotenuse 10.2 centimeters 120°. The radius bisects the angle, so the right triangl 90° triangle. SOLUTION: Find the height of the right triangle.

The apothem is .

The volume of a pyramid is , where B is the ANSWER: area of the base and h is the height of the pyramid. 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg ANSWER: of 8 centimeters and a hypotenuse of 10 centimeters. 1376.7 ft3 Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the 15. a triangular pyramid with a height of 4.8 centimeters triangle. and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for The volume of a pyramid is , where B is the the volume of a pyramid and solve for the height h. area of the base and h is the height of the pyramid.

ANSWER: Therefore, the height of the triangular pyramid is 18 3 35.6 cm cm.

16. A triangular pyramid with a right triangle base with a ANSWER: leg 8 centimeters and hypotenuse 10 centimeters has 18 cm a volume of 144 cubic centimeters. Find the height. Find the volume of each cone. Round to the SOLUTION: nearest tenth. The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle. 17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. 18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18 cm.

ANSWER: 18 cm

Find the volume of each cone. Round to the Therefore, the volume of the cone is about 134.8 nearest tenth. 3 cm .

ANSWER: 134.8 cm3 17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height 19. of the cone. SOLUTION: Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 3 235.6 in The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 1473.1 3 cone. The radius of this cone is 4.2 centimeters and cm . the height is 7.3 centimeters. ANSWER: 1473.1 cm3

Therefore, the volume of the cone is about 134.8 20. 3 cm . SOLUTION:

ANSWER: 134.8 cm3

Use trigonometric ratios to find the height h and the radius r of the cone. 19. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Use a trigonometric ratio to find the height h of the cone.

The volume of a circular cone is , where

3 r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 2.8 ft . cone. The radius of this cone is 8 centimeters. ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

Therefore, the volume of the cone is about 1473.1 The volume of a circular cone is , where 3 cm . r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, ANSWER: the radius is or 8 inches. 3 1473.1 cm

20.

SOLUTION: Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

22. a right cone with a slant height of 5.6 centimeters Use trigonometric ratios to find the height h and the and a radius of 1 centimeter radius r of the cone. SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean

Theorem to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft . ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches 3 SOLUTION: Therefore, the volume of the cone is about 5.8 cm . ANSWER: The volume of a circular cone is , where 5.8 cm3 r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, 23. SNACKS Approximately how many cubic the radius is or 8 inches. centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 1072.3 cone. Since the diameter of the cone is 8 3 in . centimeters, the radius is or 4 centimeters. The ANSWER: height of the cone is 14 centimeters.

3 1072.3 in 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean 3 Theorem to find the height h of the cone. Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

3 ANSWER: Therefore, the volume of the cone is about 5.8 cm . 3 42,000,000 ft ANSWER: 5.8 cm3 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base 23. SNACKS Approximately how many cubic has side lengths of 2 feet. What is the volume of the centimeters of roasted peanuts will completely fill a greenhouse? paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 SOLUTION: centimeters, the radius is or 4 centimeters. The The volume of a pyramid is , where B is the height of the cone is 14 centimeters. area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with

sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 3 234.6 cm Use a trigonometric ratio to find the apothem a. 24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid.

SOLUTION: The height of this pyramid is 5 feet. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: Therefore, the volume of the greenhouse is about 3 3 42,000,000 ft 32.2 ft . ANSWER: 25. GARDENING The greenhouse is a regular 3 octagonal pyramid with a height of 5 feet. The base 32.2 ft has side lengths of 2 feet. What is the volume of the greenhouse? Find the volume of each solid. Round to the nearest tenth.

26.

SOLUTION: SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

ANSWER: 471.2 in3 Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet. 27.

SOLUTION:

Therefore, the volume of the greenhouse is about ANSWER: 3 3 32.2 ft . 3190.6 m ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth. 28. SOLUTION:

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase? ANSWER: 471.2 in3

SOLUTION: 27. The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base SOLUTION: is

The volume of the ceiling is

ANSWER: 3 3190.6 m

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 28. 6666.67 × 2 = 13,333 BTUs.

SOLUTION: ANSWER: 13,333 BTUs

ANSWER: Determine the volume of the model. Explain why 3 7698.5 cm knowing the volume is helpful in this situation.

29. HEATING Sam is building an art studio in her SOLUTION: backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) The volume of a pyramid is , where B is the required to heat the building. For new construction area of the base and h is the height of the pyramid. with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

It tells Marta how much clay is needed to make the model.

SOLUTION: ANSWER: The building can be broken down into the rectangular 2 in3; It tells Marta how much clay is needed to base and the pyramid ceiling. The volume of the base make the model. is 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. The volume of the ceiling is c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

a. Double h. ANSWER: 13,333 BTUs

Determine the volume of the model. Explain why b. Double r. knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

It tells Marta how much clay is needed to make the model.

ANSWER: 3 2 in ; It tells Marta how much clay is needed to 3 make the model. volume is multiplied by 2 or 8.

31. CHANGING DIMENSIONS A cone has a radius ANSWER: of 4 centimeters and a height of 9 centimeters. a. The volume is doubled. 2 Describe how each change affects the volume of the b. The volume is multiplied by 2 or 4. cone. c. The volume is multiplied by 23 or 8. a. The height is doubled. b. The radius is doubled. Find each measure. Round to the nearest tenth c. Both the radius and the height are doubled. if necessary. 32. A square pyramid has a volume of 862.5 cubic SOLUTION: centimeters and a height of 11.5 centimeters. Find Find the volume of the original cone. Then alter the the side length of the base. values. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

a. Double h.

The volume is doubled.

b. Double r. The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? 2 The volume is multiplied by 2 or 4. SOLUTION:

c. Double r and h. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

volume is multiplied by 23 or 8.

ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find The diameter is 2(7) or 14 inches. the side length of the base. SOLUTION: ANSWER: 14 in. The volume of a pyramid is , where B is the 34. The lateral area of a cone is 71.6 square millimeters area of the base and h is the height of the pyramid. and the slant height is 6 millimeters. What is the Let s be the side length of the base. volume of the cone?

SOLUTION:

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the The side length of the base is 15 cm. radius r.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter?

SOLUTION: So, the radius is about 3.8 millimeters.

The volume of a circular cone is , or Use the Pythagorean Theorem to find the height of the cone. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

ANSWER: 14 in.

34. The lateral area of a cone is 71.6 square millimeters Therefore, the volume of the cone is about 70.2 and the slant height is 6 millimeters. What is the 3 volume of the cone? mm . SOLUTION: ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. The lateral area of a cone is , where r is c. ANALYTICAL Explain how multiplying the base the radius and is the slant height of the cone. area and/or the height of the pyramid by 5 affects the Replace L with 71.6 and with 6, then solve for the volume of the pyramid. radius r. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the b. The volumes are the same. The volume of a cone. pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this

problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with c. If the base area is multiplied by 5, the volume is different bases that have a height of 10 centimeters multiplied by 5. If the height is multiplied by 5, the and a base area of 24 square centimeters. volume is multiplied by 5. If both the base area and b. VERBAL What is true about the volumes of the the height are multiplied by 5, the volume is multiplied two pyramids that you drew? Explain. by 5 · 5 or 25. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

by 5 · 5 or 25. b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

. The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

ANSWER: a. Sample answer:

or 10h cubic units.

ANSWER: Sometimes; the statement is true if the base area of the cone is 3 times as great as the base area of the b. The volumes are the same. The volume of a prism. For example, if the base of the prism has an pyramid equals one third times the base area times area of 10 square units, then its volume is 10h cubic the height. So, if the base areas of two pyramids are units. So, the cone must have a base area of 30 equal and their heights are equal, then their volumes square units so that its volume is or 10h are equal. c. If the base area is multiplied by 5, the volume is cubic units. multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and 37. ERROR ANALYSIS Alexandra and Cornelio are the height are multiplied by 5, the volume is multiplied calculating the volume of the cone below. Is either of by 5 · 5 or 25. them correct? Explain your answer.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the SOLUTION: prism. Set the volumes equal. The slant height is used for surface area, but the height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. The volumes will only be equal when the radius of SOLUTION: the cone is equal to or when . 3 1704 cm ; The formula for the volume of a cylinder Therefore, the statement is true sometimes if the is V= Bh, while the formula for the volume of a cone base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the is V = Bh. The volume of a cylinder is three times prism has an area of 10 square units, then its volume as much as the volume of a cone with the same is 10h cubic units. So, the cone must have a base radius and height. area of 30 square units so that its volume is ANSWER: or 10h cubic units. 1704 cm3; The volume of a cylinder is three times as much as the volume of a cone with the same radius ANSWER: and height. Sometimes; the statement is true if the base area of the cone is 3 times as great as the base area of the 39. OPEN ENDED Give an example of a pyramid and prism. For example, if the base of the prism has an a prism that have the same base and the same area of 10 square units, then its volume is 10h cubic volume. Explain your reasoning. units. So, the cone must have a base area of 30 SOLUTION: square units so that its volume is or 10h The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of

cubic units. that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of 37. ERROR ANALYSIS Alexandra and Cornelio are the pyramid must be 3 times as great as the height of calculating the volume of the cone below. Is either of the prism. them correct? Explain your answer.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

SOLUTION: Sample answer: The slant height is used for surface area, but the A square pyramid with a base area of 16 and a height is used for volume. For this cone, the slant height of 12, a prism with a square base area of 16 height of 13 is provided, and we need to calculate the and a height of 4. height before we can calculate the volume. ANSWER:

Sample answer: A square pyramid with a base area Alexandra incorrectly used the slant height. of 16 and a height of 12, a prism with a square base area of 16 and a height of 4; if a pyramid and prism ANSWER: have the same base, then in order to have the same Cornelio; Alexandra incorrectly used the slant height. volume, the height of the pyramid must be 3 times as great as the height of the prism. 38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that 40. WRITING IN MATH Compare and contrast has the same radius and height as the cone? Explain finding volumes of pyramids and cones with finding your reasoning. volumes of prisms and cylinders. SOLUTION: SOLUTION: 3 To find the volume of each solid, you must know the 1704 cm ; The formula for the volume of a cylinder area of the base and the height. The volume of a is V= Bh, while the formula for the volume of a cone pyramid is one third the volume of a prism that has is V = Bh. The volume of a cylinder is three times the same height and base area. The volume of a as much as the volume of a cone with the same cone is one third the volume of a cylinder that has the radius and height. same height and base area.

ANSWER: ANSWER: 1704 cm3; The volume of a cylinder is three times as To find the volume of each solid, you must know the much as the volume of a cone with the same radius area of the base and the height. The volume of a and height. pyramid is one third the volume of a prism that has the same height and base area. The volume of a 39. OPEN ENDED Give an example of a pyramid and cone is one third the volume of a cylinder that has the a prism that have the same base and the same same height and base area. volume. Explain your reasoning. 41. A conical sand toy has the dimensions as shown SOLUTION: below. How many cubic centimeters of sand will it The formula for volume of a prism is V = Bh and the hold when it is filled to the top? formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of

the prism. A 12π

B 15π

Set the base areas of the prism and pyramid, and SOLUTION: make the height of the pyramid equal to 3 times the Use the Pythagorean Theorem to find the radius r of height of the prism. the cone.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. So, the radius of the cone is 3 centimeters. SOLUTION: The volume of a circular cone is , where To find the volume of each solid, you must know the r is the radius of the base and h is the height of the area of the base and the height. The volume of a cone. pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

ANSWER: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has Therefore, the correct choice is A. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the ANSWER: same height and base area. A

41. A conical sand toy has the dimensions as shown 42. SHORT RESPONSE Brooke is buying a tent that below. How many cubic centimeters of sand will it is in the shape of a rectangular pyramid. The base is hold when it is filled to the top? 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION:

The volume of a pyramid is , where B is the A 12π area of the base and h is the height of the pyramid. B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the radius of the cone is 3 centimeters. F

The volume of a circular cone is , where G r is the radius of the base and h is the height of the cone. H

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Therefore, the correct choice is A. Favorable outcomes: {5 orange} Number of favorable outcomes: 5 ANSWER: A

The volume of a pyramid is , where B is the ANSWER: area of the base and h is the height of the pyramid. F

44. SAT/ACT For all

A –8 B x – 4 C D E

ANSWER: SOLUTION: 5.5 ft

ANSWER: E Find the volume of each prism.

G 45.

H SOLUTION: The volume of a prism is , where B is the J area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 SOLUTION: inches and a width of 12 inches. The height h of the prism is 6 inches. Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 1008 in3 So, the correct choice is F.

ANSWER: F

46. 44. SAT/ACT For all SOLUTION: A –8 The volume of a prism is , where B is the B x – 4 area of the base and h is the height of the prism. The C base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will D bisect the base. Use the Pythagorean Theorem to E find the height of the triangle.

SOLUTION:

So, the correct choice is E.

ANSWER: E Find the volume of each prism.

So, the height of the triangle is 12 feet. Find the area of the triangle.

45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the So, the area of the base B is 60 ft2. prism is 6 inches. The height h of the prism is 19 feet.

3 3 Therefore, the volume of the prism is 1008 in . Therefore, the volume of the prism is 1140 ft .

ANSWER: ANSWER: 3 1008 in3 1140 ft

46. 47. SOLUTION: SOLUTION: The volume of a prism is , where B is the The volume of a prism is , where B is the area of the base and h is the height of the prism. The area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base base of this prism is a rectangle with a length of 79.4 of 10 feet and two legs of 13 feet. The height h will meters and a width of 52.5 meters. The height of the bisect the base. Use the Pythagorean Theorem to prism is 102.3 meters. find the height of the triangle.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain So, the height of the triangle is 12 feet. Find the area after harvest. The cone at the bottom of the bin of the triangle. allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the So, the area of the base B is 60 ft2. surface area of the conical top and bottom and find the surface area of the cylinder and add them. The height h of the prism is 19 feet. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 1140 ft

47. Find the slant height of the conical bottom. SOLUTION: The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3. The formula for finding the surface area of a cylinder ANSWER: is , where is r is the radius and h is the slant 3 426,437.6 m height of the cylinder.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin Surface area of the bin = Surface area of the allows the grain to be emptied more easily. Use the cylinder + Surface area of the conical top + surface dimensions in the diagram to find the entire surface area of the conical bottom. area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find ANSWER: the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height Find the area of each shaded region. Polygons

of the cone. in 50 - 52 are regular.

Find the slant height of the conical top.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. ANSWER: 30.4 cm2

50. SOLUTION:

Area of the shaded region = Area of the circle – Area of the hexagon

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the Surface area of the bin = Surface area of the angle, so we use 30° when using trig to find the cylinder + Surface area of the conical top + surface length of the apothem. area of the conical bottom.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Now find the area of the hexagon.

ANSWER: 30.4 cm2

50. SOLUTION: Area of the shaded region = Area of the circle – ANSWER: 2 Area of the hexagon 54.4 in

51. A regular hexagon has 6 sides, so the measure of the SOLUTION: interior angle is . The apothem bisects the The shaded area is the area of the equilateral triangle angle, so we use 30° when using trig to find the less the area of the inscribed circle. length of the apothem. Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

ANSWER: 2 54.4 in

So, the area of the equilateral triangle is about 67.3 square feet. 51. SOLUTION: Subtract to find the area of shaded region. The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 So, the area of the circle is about 40.7 square feet. 26.6 ft

Next, find the area of the equilateral triangle.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b The equilateral triangle can be divided into three represent the length of each side of the equilateral triangle and h represent the radius of the inscribed isosceles triangles with central angles of or circle. 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The Use a trigonometric ratio to find the value of b. angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and h.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find Therefore, the area of the shaded region is about 2 the area of the shaded region. 26.6 ft . A(shaded region) = A(circumscribed circle) - A ANSWER: (equilateral triangle) + A(inscribed circle) 2 26.6 ft

52. Therefore, the area of the shaded region is about SOLUTION: 168.2 mm2. The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b ANSWER: 2 represent the length of each side of the equilateral 168.2 mm triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

Find the volume of each pyramid.

1. SOLUTION:

The volume of a pyramid is , where B is the ANSWER: area of the base and h is the height of the pyramid. 3 The base of this pyramid is a right triangle with legs 132 cm of 9 inches and 5 inches and the height of the 3. a rectangular pyramid with a height of 5.2 meters pyramid is 10 inches. and a base 8 meters by 4.5 meters

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 3 75 in

ANSWER: 3 62.4 m

2. 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with area of the base and h is the height of the pyramid. sides of 4.4 centimeters and an apothem of 3 The base of this pyramid is a square with sides of 8 centimeters. The height of the pyramid is 12 meters. The height of the pyramid is 14 meters. centimeters.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the ANSWER: nearest tenth. 132 cm3

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION: 5.

The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a circular cone is , or The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of , where B is the area of the base, h is the the pyramid is 5.2 meters. height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a ANSWER: base with 8-meter side lengths 3 51.3 in SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

6. SOLUTION:

ANSWER: Use trigonometry to find the radius r.

298.7 m3

Find the volume of each cone. Round to the nearest tenth. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base.

5. The height of the cone is 11.5 centimeters.

SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. ANSWER: Since the diameter of this cone is 7 inches, the radius 168.1 cm3 is or 3.5 inches. The height of the cone is 4 inches. 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the ANSWER: height of the cone, and r is the radius of the base. 3 51.3 in The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

6. SOLUTION: ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters Use trigonometry to find the radius r. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the Use the Pythagorean Theorem to find the height h of height of the cone, and r is the radius of the base. the cone. Then find its volume. The height of the cone is 11.5 centimeters.

ANSWER: So, the height of the cone is 20 meters. 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or ANSWER: , where B is the area of the base, h is the 4712.4 m3 height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the 9. MUSEUMS The sky dome of the National Corvette height is 10.5 millimeters. Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B ANSWER: is the area of the base and h is the height of the 3 cone. 28.1 mm For this cone, the area of the base is 15,400 square 8. a cone with a slant height of 25 meters and a radius feet and the height is 100 feet. of 15 meters

SOLUTION:

ANSWER: Use the Pythagorean Theorem to find the height h of 3 the cone. Then find its volume. 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

So, the height of the cone is 20 meters. 10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 4712.4 m3

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the ANSWER: base is about 15,400 square feet, find the volume of 3 air that the heating and cooling systems would have 605 in to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. 11. For this cone, the area of the base is 15,400 square feet and the height is 100 feet. SOLUTION: The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of ANSWER: 3 each pyramid. Round to the nearest tenth if 105.8 mm necessary.

12. 10. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 482.1 m3 605 in3

13. 11. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 3 105.8 mm

12. SOLUTION: The apothem is . The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 12-5 Volumes of Pyramids and Cones 3 482.1 m3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the 13. area of the base and h is the height of the pyramid. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl ANSWER:

90° triangle. 3 1376.7 ft 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The apothem is .

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. eSolutions Manual - Powered by Cognero Page 5

ANSWER: 3 35.6 cm ANSWER: 3 16. A triangular pyramid with a right triangle base with a 1376.7 ft leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and SOLUTION: hypotenuse 10.2 centimeters The base of the pyramid is a right triangle with a leg SOLUTION: of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a Find the height of the right triangle. of the right triangle and then find the area of the triangle.

The length of the other leg of the right triangle is 6 cm.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height.

SOLUTION: Therefore, the height of the triangular pyramid is 18 The base of the pyramid is a right triangle with a leg cm. of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a ANSWER: of the right triangle and then find the area of the 18 cm triangle. Find the volume of each cone. Round to the nearest tenth.

17.

SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the The length of the other leg of the right triangle is 6 radius is or 5 inches. The height of the cone is 9 cm. inches.

2 So, the area of the base B is 24 cm . 3 Therefore, the volume of the cone is about 235.6 in .

Replace V with 144 and B with 24 in the formula for ANSWER: the volume of a pyramid and solve for the height h. 3 235.6 in

18. SOLUTION:

Therefore, the height of the triangular pyramid is 18 The volume of a circular cone is , where cm. r is the radius of the base and h is the height of the ANSWER: cone. The radius of this cone is 4.2 centimeters and

18 cm the height is 7.3 centimeters.

Find the volume of each cone. Round to the nearest tenth.

17. Therefore, the volume of the cone is about 134.8 3 SOLUTION: cm .

The volume of a circular cone is , ANSWER: where r is the radius of the base and h is the height 134.8 cm3 of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches. 19.

SOLUTION:

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 3 Use a trigonometric ratio to find the height h of the 235.6 in cone.

18. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the volume of the cone is about 1473.1 3 cm .

ANSWER: Therefore, the volume of the cone is about 134.8 1473.1 cm3 3 cm .

ANSWER: 134.8 cm3 20. SOLUTION:

19. SOLUTION:

Use trigonometric ratios to find the height h and the radius r of the cone.

Use a trigonometric ratio to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: Therefore, the volume of the cone is about 1473.1 2.8 ft3 3 cm . 21. an oblique cone with a diameter of 16 inches and an ANSWER: altitude of 16 inches 1473.1 cm3 SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches,

20. the radius is or 8 inches.

SOLUTION:

Therefore, the volume of the cone is about 1072.3 Use trigonometric ratios to find the height h and the 3 radius r of the cone. in . ANSWER: 1072.3 in3

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The volume of a circular cone is , where The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean r is the radius of the base and h is the height of the Theorem to find the height h of the cone. cone.

3 Therefore, the volume of the cone is about 2.8 ft . ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, 3 the radius is or 8 inches. Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base

diameter of 8 centimeters? Round to the nearest Therefore, the volume of the cone is about 1072.3 3 tenth. in . SOLUTION: ANSWER: The volume of a circular cone is , where 1072.3 in3 r is the radius of the base and h is the height of the 22. a right cone with a slant height of 5.6 centimeters cone. Since the diameter of the cone is 8 and a radius of 1 centimeter centimeters, the radius is or 4 centimeters. The SOLUTION: height of the cone is 14 centimeters. The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 3 234.6 cm 24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a ANSWER: 3 paper cone that is 14 centimeters high and has a base 42,000,000 ft diameter of 8 centimeters? Round to the nearest tenth. 25. GARDENING The greenhouse is a regular SOLUTION: octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the The volume of a circular cone is , where greenhouse? r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with 3 Therefore, the paper cone will hold about 234.6 cm sides of 2 feet. A central angle of the octagon is of roasted peanuts. or 45°, so the angle formed in the triangle ANSWER: below is 22.5°.

234.6 cm3

SOLUTION: Use a trigonometric ratio to find the apothem a.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The height of this pyramid is 5 feet.

ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse? Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 26. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is SOLUTION: or 45°, so the angle formed in the triangle Volume of the solid given = Volume of the small below is 22.5°. cone + Volume of the large cone

Use a trigonometric ratio to find the apothem a.

ANSWER: 471.2 in3 The height of this pyramid is 5 feet.

27. SOLUTION:

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the ANSWER: nearest tenth. 3 3190.6 m

26.

SOLUTION: 28. Volume of the solid given = Volume of the small cone + Volume of the large cone SOLUTION:

ANSWER: 7698.5 cm3

ANSWER: 29. HEATING Sam is building an art studio in her 3 backyard. To buy a heating unit for the space, she 471.2 in needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

27. SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

ANSWER: 3 3190.6 m

The volume of the ceiling is

28. SOLUTION:

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

ANSWER: 30. SCIENCE Marta is studying crystals that grow on 3 rock formations.For a project, she is making a clay 7698.5 cm model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of 29. HEATING Sam is building an art studio in her each pyramid will be 1 by 1.5 inches, and the total backyard. To buy a heating unit for the space, she height will be 4 inches. needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per Determine the volume of the model. Explain why cubic foot. What size unit does Sam need to knowing the volume is helpful in this situation. purchase? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is It tells Marta how much clay is needed to make the model.

ANSWER: 2 in3; It tells Marta how much clay is needed to

make the model. The volume of the ceiling is 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled.

c. Both the radius and the height are doubled. The total volume is therefore 5000 + 1666.67 = 3 SOLUTION: 6666.67 ft . Two BTU's are needed for every cubic Find the volume of the original cone. Then alter the foot, so the size of the heating unit Sam should buy is values. 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the The volume is doubled.

area of the base and h is the height of the pyramid. b. Double r.

It tells Marta how much clay is needed to make the 2 model. The volume is multiplied by 2 or 4.

ANSWER: c. Double r and h. 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. volume is multiplied by 23 or 8. b. The radius is doubled. c. Both the radius and the height are doubled. ANSWER: a. The volume is doubled. SOLUTION: 2 Find the volume of the original cone. Then alter the b. The volume is multiplied by 2 or 4. 3 values. c. The volume is multiplied by 2 or 8. Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

a. Double h. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The volume is doubled.

b. Double r.

The side length of the base is 15 cm. 2 The volume is multiplied by 2 or 4. ANSWER:

15 cm c. Double r and h. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

volume is multiplied by 23 or 8. , where B is the area of the base, h is the

ANSWER: height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 a. The volume is doubled. centimeters. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base. The diameter is 2(7) or 14 inches.

ANSWER:

14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The side length of the base is 15 cm.

ANSWER:

15 cm The lateral area of a cone is , where r is the radius and is the slant height of the cone. 33. The volume of a cone is 196π cubic inches and the Replace L with 71.6 and with 6, then solve for the height is 12 inches. What is the diameter? radius r. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 So, the radius is about 3.8 millimeters. centimeters. Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The diameter is 2(7) or 14 inches. The volume of a circular cone is , where ANSWER: r is the radius of the base and h is the height of the 14 in. cone.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this The lateral area of a cone is , where r is problem, you will investigate rectangular pyramids. the radius and is the slant height of the cone. a. GEOMETRIC Draw two pyramids with Replace L with 71.6 and with 6, then solve for the different bases that have a height of 10 centimeters

radius r. and a base area of 24 square centimeters.

b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: So, the radius is about 3.8 millimeters. a. Use rectangular bases and pick values that

multiply to make 24. Use the Pythagorean Theorem to find the height of the cone. Sample answer:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are Therefore, the volume of the cone is about 70.2 equal and their heights are equal, then their volumes 3 are equal. mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base

area and/or the height of the pyramid by 5 affects the volume of the pyramid. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the SOLUTION: volume is multiplied by 5. If both the base area and a. Use rectangular bases and pick values that the height are multiplied by 5, the volume is multiplied multiply to make 24. by 5 · 5 or 25.

Sample answer:

ANSWER: a. Sample answer:

The volumes will only be equal when the radius of the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is b. The volumes are the same. The volume of a pyramid equals one third times the base area times or 10h cubic units. the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes ANSWER: are equal. Sometimes; the statement is true if the base area of c. If the base area is multiplied by 5, the volume is the cone is 3 times as great as the base area of the multiplied by 5. If the height is multiplied by 5, the prism. For example, if the base of the prism has an volume is multiplied by 5. If both the base area and area of 10 square units, then its volume is 10h cubic the height are multiplied by 5, the volume is multiplied units. So, the cone must have a base area of 30 by 5 · 5 or 25. square units so that its volume is or 10h CCSS ARGUMENTS 36. Determine whether the cubic units. following statement is sometimes, always, or never true. Justify your reasoning. 37. ERROR ANALYSIS Alexandra and Cornelio are The volume of a cone with radius r and height h calculating the volume of the cone below. Is either of equals the volume of a prism with height h. them correct? Explain your answer. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

Alexandra incorrectly used the slant height.

The volumes will only be equal when the radius of ANSWER: Cornelio; Alexandra incorrectly used the slant height. the cone is equal to or when . Therefore, the statement is true sometimes if the 38. REASONING A cone has a volume of 568 cubic base area of the cone is 3 times as great as the base centimeters. What is the volume of a cylinder that area of the prism. For example, if the base of the has the same radius and height as the cone? Explain prism has an area of 10 square units, then its volume your reasoning. is 10h cubic units. So, the cone must have a base SOLUTION: area of 30 square units so that its volume is 3 1704 cm ; The formula for the volume of a cylinder or 10h cubic units. is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times ANSWER: as much as the volume of a cone with the same Sometimes; the statement is true if the base area of radius and height. the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an ANSWER: area of 10 square units, then its volume is 10h cubic 3 1704 cm ; The volume of a cylinder is three times as units. So, the cone must have a base area of 30 much as the volume of a cone with the same radius square units so that its volume is or 10h and height. cubic units. 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same 37. ERROR ANALYSIS Alexandra and Cornelio are volume. Explain your reasoning. calculating the volume of the cone below. Is either of them correct? Explain your answer. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

SOLUTION: The slant height is used for surface area, but the height is used for volume. For this cone, the slant Set the base areas of the prism and pyramid, and height of 13 is provided, and we need to calculate the make the height of the pyramid equal to 3 times the height before we can calculate the volume. height of the prism.

Sample answer: Alexandra incorrectly used the slant height. A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 ANSWER: and a height of 4. Cornelio; Alexandra incorrectly used the slant height. ANSWER: 38. REASONING A cone has a volume of 568 cubic Sample answer: A square pyramid with a base area centimeters. What is the volume of a cylinder that of 16 and a height of 12, a prism with a square base has the same radius and height as the cone? Explain area of 16 and a height of 4; if a pyramid and prism your reasoning. have the same base, then in order to have the same SOLUTION: volume, the height of the pyramid must be 3 times as 3 great as the height of the prism. 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone 40. WRITING IN MATH Compare and contrast is V = Bh. The volume of a cylinder is three times finding volumes of pyramids and cones with finding as much as the volume of a cone with the same volumes of prisms and cylinders. radius and height. SOLUTION: ANSWER: To find the volume of each solid, you must know the 3 area of the base and the height. The volume of a 1704 cm ; The volume of a cylinder is three times as pyramid is one third the volume of a prism that has much as the volume of a cone with the same radius the same height and base area. The volume of a and height. cone is one third the volume of a cylinder that has the same height and base area. 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same ANSWER: volume. Explain your reasoning. To find the volume of each solid, you must know the SOLUTION: area of the base and the height. The volume of a pyramid is one third the volume of a prism that has The formula for volume of a prism is V = Bh and the the same height and base area. The volume of a formula for the volume of a pyramid is one-third of cone is one third the volume of a cylinder that has the that. So, if a pyramid and prism have the same base, same height and base area. then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of 41. A conical sand toy has the dimensions as shown the prism. below. How many cubic centimeters of sand will it hold when it is filled to the top?

A 12π Set the base areas of the prism and pyramid, and B 15π make the height of the pyramid equal to 3 times the height of the prism. C

Sample answer: D A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 SOLUTION: and a height of 4. Use the Pythagorean Theorem to find the radius r of

ANSWER: the cone. Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a So, the radius of the cone is 3 centimeters. cone is one third the volume of a cylinder that has the same height and base area. The volume of a circular cone is , where ANSWER: r is the radius of the base and h is the height of the To find the volume of each solid, you must know the cone. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top? Therefore, the correct choice is A. ANSWER: A

42. SHORT RESPONSE Brooke is buying a tent that A 12π is in the shape of a rectangular pyramid. The base is B 15π 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? C SOLUTION:

D The volume of a pyramid is , where B is the

SOLUTION: area of the base and h is the height of the pyramid. Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange? So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the correct choice is A. J

ANSWER: SOLUTION: A Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} SHORT RESPONSE 42. Brooke is buying a tent that Number of possible outcomes : 25 is in the shape of a rectangular pyramid. The base is Favorable outcomes: {5 orange} 6 feet by 8 feet. If the tent holds 88 cubic feet of air, Number of favorable outcomes: 5

how tall is the tent’s center pole? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the correct choice is F.

ANSWER: F

44. SAT/ACT For all

A –8 ANSWER: B x – 4 5.5 ft C D 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows E the results of several spins. What is the experimental probability of the spinner landing on orange? SOLUTION:

F So, the correct choice is E. ANSWER: G E

H Find the volume of each prism.

SOLUTION: 45. Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} SOLUTION: Number of possible outcomes : 25 The volume of a prism is , where B is the Favorable outcomes: {5 orange} area of the base and h is the height of the prism. The Number of favorable outcomes: 5 base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the correct choice is F. 3 ANSWER: Therefore, the volume of the prism is 1008 in . F ANSWER: 1008 in3 44. SAT/ACT For all

A –8 B x – 4 C D 46. E SOLUTION: The volume of a prism is , where B is the SOLUTION: area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the correct choice is E.

ANSWER: E

Find the volume of each prism.

45.

SOLUTION: The volume of a prism is , where B is the So, the height of the triangle is 12 feet. Find the area area of the base and h is the height of the prism. The of the triangle. base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the area of the base B is 60 ft2.

3 The height h of the prism is 19 feet. Therefore, the volume of the prism is 1008 in . ANSWER: 1008 in3

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 46. 1140 ft SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to 47. find the height of the triangle. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3. So, the height of the triangle is 12 feet. Find the area of the triangle. ANSWER: 3 426,437.6 m

SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is 3 Therefore, the volume of the prism is 1140 ft . , where is r is the radius and l is the slant height of the cone. ANSWER: 3 1140 ft Find the slant height of the conical top.

47. SOLUTION:

The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters. Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin The formula for finding the surface area of a cylinder allows the grain to be emptied more easily. Use the is , where is r is the radius and h is the slant dimensions in the diagram to find the entire surface height of the cylinder. area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. Surface area of the bin = Surface area of the SOLUTION: cylinder + Surface area of the conical top + surface To find the entire surface area of the bin, find the area of the conical bottom. surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top. ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49.

SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Find the slant height of the conical bottom. Area of the rectangle = 10(5) The height of the conical bottom is 28 – (5 + 12 + 2) = 50 or 9 ft.

ANSWER: 30.4 cm2

The formula for finding the surface area of a cylinder 50. is , where is r is the radius and h is the slant SOLUTION: height of the cylinder. Area of the shaded region = Area of the circle – Area of the hexagon

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: 2 30.4 cm Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

ANSWER: 2 A regular hexagon has 6 sides, so the measure of the 54.4 in interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem. 51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three

isosceles triangles with central angles of or Now find the area of the hexagon. 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

ANSWER: 2 54.4 in

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. So, the area of the equilateral triangle is about 67.3 Find the area of the circle with a radius of 3.6 feet. square feet.

Subtract to find the area of shaded region.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle. Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft

The equilateral triangle can be divided into three 52. isosceles triangles with central angles of or 120°, so the angle in the triangle created by the SOLUTION: height of the isosceles triangle is 60° and the base is The area of the shaded region is the area of the half the length b of the side of the equilateral triangle. circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Use a trigonometric ratio to find the value of b.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The

Now use the area of the isosceles triangles to find angle in the triangle formed by the height.h of the the area of the equilateral triangle. isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

So, the area of the equilateral triangle is about 67.3 square feet. Use trigonometric ratios to find the values of b and Subtract to find the area of shaded region. h.

Therefore, the area of the shaded region is about

26.6 ft2.

ANSWER: 2 26.6 ft The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A

52. (equilateral triangle) + A(inscribed circle) SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed Therefore, the area of the shaded region is about 2 circle. 168.2 mm .

ANSWER: 2 168.2 mm

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

Find the volume of each pyramid.

ANSWER: 132 cm3

1. 3. a rectangular pyramid with a height of 5.2 meters SOLUTION: and a base 8 meters by 4.5 meters SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a right triangle with legs area of the base and h is the height of the pyramid. of 9 inches and 5 inches and the height of the The base of this pyramid is a rectangle with a length pyramid is 10 inches. of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 3 ANSWER: 62.4 m 3 75 in 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 2. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. SOLUTION:

3 298.7 m Find the volume of each cone. Round to the nearest tenth.

5. ANSWER: SOLUTION: 132 cm3 The volume of a circular cone is , or 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius The volume of a pyramid is , where B is the is or 3.5 inches. The height of the cone is 4 inches. area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 3 51.3 in

ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a

base with 8-meter side lengths 6. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

Use trigonometry to find the radius r.

The volume of a circular cone is , or

ANSWER: , where B is the area of the base, h is the 298.7 m3 height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. Find the volume of each cone. Round to the nearest tenth.

5. ANSWER: SOLUTION: 168.1 cm3

The volume of a circular cone is , or 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius The volume of a circular cone is , or is or 3.5 inches. The height of the cone is 4 inches. , where B is the area of the base, h is the

height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

ANSWER: 3 51.3 in

ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius 6. of 15 meters SOLUTION: SOLUTION:

Use trigonometry to find the radius r. Use the Pythagorean Theorem to find the height h of

the cone. Then find its volume.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. So, the height of the cone is 20 meters.

ANSWER: 3 168.1 cm ANSWER: 7. an oblique cone with a height of 10.5 millimeters and 4712.4 m3 a radius of 1.6 millimeters 9. MUSEUMS The sky dome of the National Corvette SOLUTION: Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the The volume of a circular cone is , or base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have , where B is the area of the base, h is the to accommodate. Round to the nearest tenth. height of the cone, and r is the radius of the base. SOLUTION: The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. The volume of a circular cone is , where B

is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters ANSWER: SOLUTION: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. So, the height of the cone is 20 meters.

ANSWER: ANSWER: 605 in3 4712.4 m3

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have 11. to accommodate. Round to the nearest tenth. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a circular cone is , where B area of the base and h is the height of the pyramid. is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 3 105.8 mm

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of 12. each pyramid. Round to the nearest tenth if necessary. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 482.1 m3

ANSWER: 13. 605 in3 SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri 11. determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl SOLUTION: 90° triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 105.8 mm The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION: ANSWER: The volume of a pyramid is , where B is the 482.1 m3 area of the base and h is the height of the pyramid.

13. SOLUTION: ANSWER: The volume of a pyramid is , where B is the 1376.7 ft3 base and h is the height of the pyramid. 15. a triangular pyramid with a height of 4.8 centimeters

and a right triangle base with a leg 5 centimeters and The base is a hexagon, so we need to make a right tri hypotenuse 10.2 centimeters determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl SOLUTION: 90° triangle. Find the height of the right triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the ANSWER: area of the base and h is the height of the pyramid. 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. ANSWER: 12-5 Volumes of Pyramids and Cones Use the Pythagorean Theorem to find the other leg a 1376.7 ft3 of the right triangle and then find the area of the triangle. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for The volume of a pyramid is , where B is the the volume of a pyramid and solve for the height h.

area of the base and h is the height of the pyramid.

ANSWER: Therefore, the height of the triangular pyramid is 18 3 cm. 35.6 cm ANSWER: 16. A triangular pyramid with a right triangle base with a 18 cm leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. Find the volume of each cone. Round to the SOLUTION: nearest tenth. The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the

triangle. 17. eSolutions Manual - Powered by Cognero SOLUTION: Page 6

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. 18.

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18 cm.

ANSWER: 18 cm Therefore, the volume of the cone is about 134.8 Find the volume of each cone. Round to the 3 nearest tenth. cm . ANSWER: 134.8 cm3

17. SOLUTION:

The volume of a circular cone is , 19. where r is the radius of the base and h is the height of the cone. SOLUTION:

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: The volume of a circular cone is , where 235.6 in3 r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

18. SOLUTION:

The volume of a circular cone is , where Therefore, the volume of the cone is about 1473.1 r is the radius of the base and h is the height of the 3 cm . cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. ANSWER:

3 1473.1 cm

20. Therefore, the volume of the cone is about 134.8 SOLUTION: 3 cm .

ANSWER: 134.8 cm3

Use trigonometric ratios to find the height h and the radius r of the cone.

19. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Use a trigonometric ratio to find the height h of the

cone.

The volume of a circular cone is , where 3 Therefore, the volume of the cone is about 2.8 ft . r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. ANSWER:

3 2.8 ft 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where Therefore, the volume of the cone is about 1473.1 3 r is the radius of the base and h is the height of the cm . cone. Since the diameter of this cone is 16 inches, ANSWER: the radius is or 8 inches. 1473.1 cm3

20. SOLUTION: Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter Use trigonometric ratios to find the height h and the radius r of the cone. SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an 3 altitude of 16 inches Therefore, the volume of the cone is about 5.8 cm . SOLUTION: ANSWER: The volume of a circular cone is , where 5.8 cm3 r is the radius of the base and h is the height of the 23. SNACKS Approximately how many cubic cone. Since the diameter of this cone is 16 inches, centimeters of roasted peanuts will completely fill a

the radius is or 8 inches. paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 Therefore, the volume of the cone is about 1072.3 3 in . centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters. ANSWER: 1072.3 in3

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant 3 height of 5.6 centimeters. Use the Pythagorean Therefore, the paper cone will hold about 234.6 cm Theorem to find the height h of the cone. of roasted peanuts.

ANSWER: 234.6 cm3

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 3 Therefore, the volume of the cone is about 5.8 cm . 42,000,000 ft ANSWER: 25. GARDENING The greenhouse is a regular 5.8 cm3 octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the 23. SNACKS Approximately how many cubic greenhouse? centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the SOLUTION: cone. Since the diameter of the cone is 8 The volume of a pyramid is , where B is the centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters. area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3 Use a trigonometric ratio to find the apothem a. 24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION: The height of this pyramid is 5 feet.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the volume of the greenhouse is about ANSWER: 3 32.2 ft . 3 42,000,000 ft ANSWER: 3 25. GARDENING The greenhouse is a regular 32.2 ft octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the Find the volume of each solid. Round to the greenhouse? nearest tenth.

26. SOLUTION: Volume of the solid given = Volume of the small SOLUTION: cone + Volume of the large cone The volume of a pyramid is , where B is the

ANSWER: 3 471.2 in Use a trigonometric ratio to find the apothem a.

27. The height of this pyramid is 5 feet. SOLUTION:

ANSWER: Therefore, the volume of the greenhouse is about 3 3190.6 m 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth. 28. SOLUTION:

26. SOLUTION: Volume of the solid given = Volume of the small

cone + Volume of the large cone

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase? ANSWER: 471.2 in3

SOLUTION: The building can be broken down into the rectangular 27. base and the pyramid ceiling. The volume of the base is SOLUTION:

The volume of the ceiling is

ANSWER: 3 3190.6 m

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. 28.

SOLUTION: ANSWER: 13,333 BTUs

ANSWER: Determine the volume of the model. Explain why knowing the volume is helpful in this situation. 7698.5 cm3 SOLUTION: 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she The volume of a pyramid is , where B is the needs to determine the BTUs (British Thermal Units) area of the base and h is the height of the pyramid. required to heat the building. For new construction

with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

It tells Marta how much clay is needed to make the model.

ANSWER: SOLUTION: 2 in3; It tells Marta how much clay is needed to The building can be broken down into the rectangular make the model. base and the pyramid ceiling. The volume of the base is 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled.

The volume of the ceiling is c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is

6666.67 × 2 = 13,333 BTUs.

a. Double h. ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total The volume is doubled. height will be 4 inches. b. Double r. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. 2 The volume is multiplied by 2 or 4.

c. Double r and h.

It tells Marta how much clay is needed to make the model.

ANSWER: 3 2 in ; It tells Marta how much clay is needed to volume is multiplied by 23 or 8. make the model. ANSWER: 31. CHANGING DIMENSIONS A cone has a radius a. The volume is doubled. of 4 centimeters and a height of 9 centimeters. 2 b. The volume is multiplied by 2 or 4. Describe how each change affects the volume of the 3 cone. c. The volume is multiplied by 2 or 8. a. The height is doubled. Find each measure. Round to the nearest tenth b. The radius is doubled. if necessary. c. Both the radius and the height are doubled. 32. A square pyramid has a volume of 862.5 cubic SOLUTION: centimeters and a height of 11.5 centimeters. Find the side length of the base. Find the volume of the original cone. Then alter the values. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

a. Double h.

The volume is doubled.

b. Double r. The side length of the base is 15 cm. ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter?

2 SOLUTION: The volume is multiplied by 2 or 4. The volume of a circular cone is , or c. Double r and h. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

volume is multiplied by 23 or 8.

ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find The diameter is 2(7) or 14 inches. the side length of the base. ANSWER: SOLUTION: 14 in. The volume of a pyramid is , where B is the 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the area of the base and h is the height of the pyramid. volume of the cone? Let s be the side length of the base. SOLUTION:

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r. The side length of the base is 15 cm. ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? So, the radius is about 3.8 millimeters. SOLUTION: Use the Pythagorean Theorem to find the height of The volume of a circular cone is , or the cone.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters Therefore, the volume of the cone is about 70.2 3 and the slant height is 6 millimeters. What is the mm . volume of the cone? ANSWER: SOLUTION: 70.2 mm3

c. ANALYTICAL Explain how multiplying the base The lateral area of a cone is , where r is area and/or the height of the pyramid by 5 affects the the radius and is the slant height of the cone. volume of the pyramid. Replace L with 71.6 and with 6, then solve for the radius r. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where b. The volumes are the same. The volume of a r is the radius of the base and h is the height of the pyramid equals one third times the base area times cone. the height. So, if the base areas of two pyramids are

equal and their heights are equal, then their volumes are equal.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with c. If the base area is multiplied by 5, the volume is different bases that have a height of 10 centimeters multiplied by 5. If the height is multiplied by 5, the and a base area of 24 square centimeters. volume is multiplied by 5. If both the base area and b. VERBAL What is true about the volumes of the the height are multiplied by 5, the volume is multiplied

two pyramids that you drew? Explain. by 5 · 5 or 25. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are ANSWER: equal and their heights are equal, then their volumes a. Sample answer: are equal.

c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. 36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

ANSWER: a. Sample answer:

or 10h cubic units.

ANSWER: Sometimes; the statement is true if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an b. The volumes are the same. The volume of a area of 10 square units, then its volume is 10h cubic pyramid equals one third times the base area times units. So, the cone must have a base area of 30 the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes square units so that its volume is or 10h are equal. cubic units. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the 37. ERROR ANALYSIS Alexandra and Cornelio are volume is multiplied by 5. If both the base area and calculating the volume of the cone below. Is either of the height are multiplied by 5, the volume is multiplied them correct? Explain your answer. by 5 · 5 or 25.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of SOLUTION: h is where B is the area of the base of the The slant height is used for surface area, but the prism. Set the volumes equal. height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain

your reasoning. The volumes will only be equal when the radius of SOLUTION: 3 the cone is equal to or when . 1704 cm ; The formula for the volume of a cylinder Therefore, the statement is true sometimes if the is V= Bh, while the formula for the volume of a cone base area of the cone is 3 times as great as the base is V = Bh. The volume of a cylinder is three times area of the prism. For example, if the base of the as much as the volume of a cone with the same prism has an area of 10 square units, then its volume radius and height. is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is ANSWER: 3 or 10h cubic units. 1704 cm ; The volume of a cylinder is three times as much as the volume of a cone with the same radius and height. ANSWER: Sometimes; the statement is true if the base area of 39. OPEN ENDED Give an example of a pyramid and the cone is 3 times as great as the base area of the a prism that have the same base and the same prism. For example, if the base of the prism has an volume. Explain your reasoning. area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 SOLUTION: The formula for volume of a prism is V = Bh and the square units so that its volume is or 10h formula for the volume of a pyramid is one-third of cubic units. that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of 37. ERROR ANALYSIS Alexandra and Cornelio are the pyramid must be 3 times as great as the height of calculating the volume of the cone below. Is either of the prism. them correct? Explain your answer.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

Sample answer: SOLUTION: A square pyramid with a base area of 16 and a The slant height is used for surface area, but the height of 12, a prism with a square base area of 16 height is used for volume. For this cone, the slant and a height of 4. height of 13 is provided, and we need to calculate the height before we can calculate the volume. ANSWER: Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base Alexandra incorrectly used the slant height. area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same ANSWER: volume, the height of the pyramid must be 3 times as Cornelio; Alexandra incorrectly used the slant height. great as the height of the prism.

38. REASONING A cone has a volume of 568 cubic 40. WRITING IN MATH Compare and contrast centimeters. What is the volume of a cylinder that finding volumes of pyramids and cones with finding has the same radius and height as the cone? Explain volumes of prisms and cylinders. your reasoning. SOLUTION: SOLUTION: To find the volume of each solid, you must know the 3 1704 cm ; The formula for the volume of a cylinder area of the base and the height. The volume of a is V= Bh, while the formula for the volume of a cone pyramid is one third the volume of a prism that has the same height and base area. The volume of a is V = Bh. The volume of a cylinder is three times cone is one third the volume of a cylinder that has the as much as the volume of a cone with the same same height and base area. radius and height. ANSWER: ANSWER: To find the volume of each solid, you must know the 1704 cm3; The volume of a cylinder is three times as area of the base and the height. The volume of a much as the volume of a cone with the same radius pyramid is one third the volume of a prism that has and height. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the 39. OPEN ENDED Give an example of a pyramid and same height and base area. a prism that have the same base and the same volume. Explain your reasoning. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it SOLUTION: hold when it is filled to the top? The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism. A 12π B 15π

SOLUTION: Set the base areas of the prism and pyramid, and Use the Pythagorean Theorem to find the radius r of make the height of the pyramid equal to 3 times the the cone. height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding So, the radius of the cone is 3 centimeters. volumes of prisms and cylinders. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the To find the volume of each solid, you must know the cone. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

ANSWER: To find the volume of each solid, you must know the area of the base and the height. The volume of a Therefore, the correct choice is A. pyramid is one third the volume of a prism that has the same height and base area. The volume of a ANSWER: cone is one third the volume of a cylinder that has the A same height and base area. 42. SHORT RESPONSE Brooke is buying a tent that 41. A conical sand toy has the dimensions as shown is in the shape of a rectangular pyramid. The base is below. How many cubic centimeters of sand will it 6 feet by 8 feet. If the tent holds 88 cubic feet of air,

hold when it is filled to the top? how tall is the tent’s center pole? SOLUTION:

The volume of a pyramid is , where B is the

A 12π area of the base and h is the height of the pyramid. B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

F So, the radius of the cone is 3 centimeters.

G The volume of a circular cone is , where

r is the radius of the base and h is the height of the H cone.

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Therefore, the correct choice is A. Number of favorable outcomes: 5

ANSWER: A

The volume of a pyramid is , where B is the ANSWER: F area of the base and h is the height of the pyramid.

44. SAT/ACT For all A –8 B x – 4 C D E

SOLUTION: ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental So, the correct choice is E. probability of the spinner landing on orange? ANSWER: E Find the volume of each prism.

G 45. SOLUTION: H The volume of a prism is , where B is the area of the base and h is the height of the prism. The J base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the SOLUTION: prism is 6 inches. Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 1008 in3

So, the correct choice is F.

ANSWER: F 46. 44. SAT/ACT For all SOLUTION: A –8 The volume of a prism is , where B is the area of the base and h is the height of the prism. The B x – 4 base of this prism is an isosceles triangle with a base C of 10 feet and two legs of 13 feet. The height h will D bisect the base. Use the Pythagorean Theorem to find the height of the triangle. E

SOLUTION:

So, the correct choice is E.

ANSWER: E Find the volume of each prism. So, the height of the triangle is 12 feet. Find the area of the triangle.

45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 So, the area of the base B is 60 ft2. inches and a width of 12 inches. The height h of the prism is 6 inches. The height h of the prism is 19 feet.

3 3 Therefore, the volume of the prism is 1140 ft . Therefore, the volume of the prism is 1008 in . ANSWER: ANSWER: 3 1140 ft 1008 in3

47. 46. SOLUTION: SOLUTION: The volume of a prism is , where B is the The volume of a prism is , where B is the area of the base and h is the height of the prism. The area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 base of this prism is an isosceles triangle with a base meters and a width of 52.5 meters. The height of the of 10 feet and two legs of 13 feet. The height h will prism is 102.3 meters. bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

Therefore, the volume of the prism is about 426,437.6 3 m . ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin So, the height of the triangle is 12 feet. Find the area allows the grain to be emptied more easily. Use the of the triangle. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find 2 So, the area of the base B is 60 ft . the surface area of the cylinder and add them. The formula for finding the surface area of a cone is The height h of the prism is 19 feet. , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 1140 ft

Find the slant height of the conical bottom. 47. The height of the conical bottom is 28 – (5 + 12 + 2) SOLUTION: or 9 ft. The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 3 m . The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant ANSWER: height of the cylinder. 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain Surface area of the bin = Surface area of the after harvest. The cone at the bottom of the bin cylinder + Surface area of the conical top + surface allows the grain to be emptied more easily. Use the area of the conical bottom. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the ANSWER: surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height Find the area of each shaded region. Polygons of the cone. in 50 - 52 are regular.

Find the slant height of the conical top. 49.

SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) ANSWER:

or 9 ft. 2 30.4 cm

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder. A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the Surface area of the bin = Surface area of the length of the apothem. cylinder + Surface area of the conical top + surface area of the conical bottom.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Now find the area of the hexagon.

ANSWER: 30.4 cm2

50. SOLUTION: ANSWER: 2 Area of the shaded region = Area of the circle – 54.4 in Area of the hexagon

51.

A regular hexagon has 6 sides, so the measure of the SOLUTION: The shaded area is the area of the equilateral triangle interior angle is . The apothem bisects the less the area of the inscribed circle. angle, so we use 30° when using trig to find the length of the apothem. Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

ANSWER: 2 54.4 in

So, the area of the equilateral triangle is about 67.3 square feet.

51. Subtract to find the area of shaded region. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

isosceles triangles with central angles of or circle. 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The

Use a trigonometric ratio to find the value of b. angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and h.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find Therefore, the area of the shaded region is about the area of the shaded region.

26.6 ft2. A(shaded region) = A(circumscribed circle) - A ANSWER: (equilateral triangle) + A(inscribed circle) 2 26.6 ft

52. Therefore, the area of the shaded region is about 168.2 mm2. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral ANSWER: triangle plus the area of the inscribed circle. Let b 2 represent the length of each side of the equilateral 168.2 mm triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

Find the volume of each pyramid.

2. SOLUTION: 1. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 area of the base and h is the height of the pyramid. centimeters. The height of the pyramid is 12 The base of this pyramid is a right triangle with legs centimeters. of 9 inches and 5 inches and the height of the pyramid is 10 inches.

ANSWER: 3 ANSWER: 132 cm 3 75 in 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length 2. of 8 meters and a width of 4.5 meters. The height of SOLUTION: the pyramid is 5.2 meters.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: The base of this pyramid is a square with sides of 8 3 meters. The height of the pyramid is 14 meters. 132 cm 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. ANSWER: The base of this pyramid is a rectangle with a length 3 of 8 meters and a width of 4.5 meters. The height of 298.7 m the pyramid is 5.2 meters. Find the volume of each cone. Round to the nearest tenth.

5. ANSWER: SOLUTION: 3 62.4 m The volume of a circular cone is , or 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius The volume of a pyramid is , where B is the is or 3.5 inches. The height of the cone is 4 inches. area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

ANSWER: 3 51.3 in ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth. 6. SOLUTION:

5. SOLUTION:

The volume of a circular cone is , or Use trigonometry to find the radius r.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius The volume of a circular cone is , or is or 3.5 inches. The height of the cone is 4 inches. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

ANSWER: 3 51.3 in ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters

6. SOLUTION: SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. Use trigonometry to find the radius r.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. ANSWER: 3 The height of the cone is 11.5 centimeters. 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. So, the height of the cone is 20 meters.

ANSWER: ANSWER: 3 28.1 mm 4712.4 m3

8. a cone with a slant height of 25 meters and a radius 9. MUSEUMS The sky dome of the National Corvette of 15 meters Museum in Bowling Green, Kentucky, is a conical SOLUTION: building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B Use the Pythagorean Theorem to find the height h of is the area of the base and h is the height of the the cone. Then find its volume. cone.

For this cone, the area of the base is 15,400 square

feet and the height is 100 feet.

So, the height of the cone is 20 meters.

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. ANSWER: 4712.4 m3

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the 10. base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have SOLUTION: to accommodate. Round to the nearest tenth. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a circular cone is , where B

is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 605 in3

ANSWER: 3 513,333.3 ft 11. CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if SOLUTION: necessary. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 105.8 mm

12. ANSWER: SOLUTION: 605 in3 The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

11. SOLUTION:

The volume of a pyramid is , where B is the ANSWER: area of the base and h is the height of the pyramid. 482.1 m3

13. ANSWER: SOLUTION: 3 105.8 mm The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 12. 120°. The radius bisects the angle, so the right triangl 90° triangle. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 482.1 m The apothem is .

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid. ANSWER: 3 233.8 cm The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 14. a pentagonal pyramid with a base area of 590 square 120°. The radius bisects the angle, so the right triangl feet and an altitude of 7 feet 90° triangle. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 1376.7 ft3 The apothem is . 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: The volume of a pyramid is , where B is the 1376.7 ft3 area of the base and h is the height of the pyramid. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

ANSWER: 3 35.6 cm

of the right triangle and then find the area of the triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The length of the other leg of the right triangle is 6 cm.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. 2 SOLUTION: So, the area of the base B is 24 cm . The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Replace V with 144 and B with 24 in the formula for Use the Pythagorean Theorem to find the other leg a the volume of a pyramid and solve for the height h. of the right triangle and then find the area of the triangle.

Therefore, the height of the triangular pyramid is 18 cm.

ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth. The length of the other leg of the right triangle is 6 cm.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height 2 So, the area of the base B is 24 cm . of the cone.

Replace V with 144 and B with 24 in the formula for Since the diameter of this cone is 10 inches, the

the volume of a pyramid and solve for the height h. radius is or 5 inches. The height of the cone is 9

inches.

3 Therefore, the height of the triangular pyramid is 18 Therefore, the volume of the cone is about 235.6 in . cm. ANSWER: ANSWER: 12-5 Volumes of Pyramids and Cones 235.6 in3 18 cm Find the volume of each cone. Round to the nearest tenth.

18.

17. SOLUTION: SOLUTION: The volume of a circular cone is , where

The volume of a circular cone is , r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and where r is the radius of the base and h is the height the height is 7.3 centimeters. of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 134.8 cm3 3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3

19. SOLUTION:

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and Use a trigonometric ratio to find the height h of the the height is 7.3 centimeters. cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 134.8 cone. The radius of this cone is 8 centimeters. 3 eSolutionscm .Manual - Powered by Cognero Page 7

ANSWER: 134.8 cm3

Therefore, the volume of the cone is about 1473.1 3 cm . 19. ANSWER: SOLUTION: 1473.1 cm3

20. SOLUTION: Use a trigonometric ratio to find the height h of the cone.

The volume of a circular cone is , where Use trigonometric ratios to find the height h and the radius r of the cone. r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 1473.1 cone. 3 cm .

ANSWER: 1473.1 cm3

3 20. Therefore, the volume of the cone is about 2.8 ft . SOLUTION: ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

Use trigonometric ratios to find the height h and the The volume of a circular cone is , where radius r of the cone. r is the radius of the base and h is the height of the

cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Therefore, the volume of the cone is about 1072.3 3 in . ANSWER: 1072.3 in3

22. a right cone with a slant height of 5.6 centimeters

and a radius of 1 centimeter

3 Therefore, the volume of the cone is about 2.8 ft . SOLUTION: The cone has a radius r of 1 centimeter and a slant ANSWER: height of 5.6 centimeters. Use the Pythagorean 3 2.8 ft Theorem to find the height h of the cone.

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches,

the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 3 3 1072.3 in Therefore, the volume of the cone is about 5.8 cm .

22. a right cone with a slant height of 5.6 centimeters ANSWER: and a radius of 1 centimeter 5.8 cm3 SOLUTION: The cone has a radius r of 1 centimeter and a slant 23. SNACKS Approximately how many cubic height of 5.6 centimeters. Use the Pythagorean centimeters of roasted peanuts will completely fill a Theorem to find the height h of the cone. paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8

centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 3 3 Therefore, the volume of the cone is about 5.8 cm . 234.6 cm

ANSWER: 24. CCSS MODELING The Pyramid Arena in 3 Memphis, Tennessee, is the third largest pyramid in 5.8 cm the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this 23. SNACKS Approximately how many cubic pyramid. centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base SOLUTION: diameter of 8 centimeters? Round to the nearest tenth. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid.

ANSWER:

3 42,000,000 ft

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse? 3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

CCSS MODELING 24. The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its SOLUTION: square base is 600 feet wide. Find the volume of this pyramid. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with The volume of a pyramid is , where B is the sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle area of the base and h is the height of the pyramid. below is 22.5°.

ANSWER: 3 42,000,000 ft Use a trigonometric ratio to find the apothem a.

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

The height of this pyramid is 5 feet.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with Therefore, the volume of the greenhouse is about sides of 2 feet. A central angle of the octagon is 32.2 ft3. or 45°, so the angle formed in the triangle ANSWER: below is 22.5°. 3 32.2 ft Find the volume of each solid. Round to the nearest tenth.

Use a trigonometric ratio to find the apothem a. 26.

SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

The height of this pyramid is 5 feet.

ANSWER: Therefore, the volume of the greenhouse is about 3 3 471.2 in 32.2 ft .

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth. 27. SOLUTION:

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 3 3190.6 m

28. SOLUTION: ANSWER: 471.2 in3

27. ANSWER: SOLUTION: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase? ANSWER: 3 3190.6 m

SOLUTION: 28. The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base SOLUTION: is

The volume of the ceiling is

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she The total volume is therefore 5000 + 1666.67 = 3 needs to determine the BTUs (British Thermal Units) 6666.67 ft . Two BTU's are needed for every cubic required to heat the building. For new construction foot, so the size of the heating unit Sam should buy is with good insulation, there should be 2 BTUs per 6666.67 × 2 = 13,333 BTUs. cubic foot. What size unit does Sam need to purchase? ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of SOLUTION: each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches. The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The volume of the ceiling is

It tells Marta how much clay is needed to make the The total volume is therefore 5000 + 1666.67 = model. 3 6666.67 ft . Two BTU's are needed for every cubic ANSWER: foot, so the size of the heating unit Sam should buy is 3 6666.67 × 2 = 13,333 BTUs. 2 in ; It tells Marta how much clay is needed to make the model.

ANSWER: 31. CHANGING DIMENSIONS A cone has a radius 13,333 BTUs of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the 30. SCIENCE Marta is studying crystals that grow on cone. rock formations.For a project, she is making a clay a. The height is doubled. model of a crystal with a shape that is a composite of b. The radius is doubled. two congruent rectangular pyramids. The base of c. Both the radius and the height are doubled. each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches. SOLUTION: Find the volume of the original cone. Then alter the Determine the volume of the model. Explain why values. knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

a. Double h.

It tells Marta how much clay is needed to make the model.

ANSWER: The volume is doubled. 3 2 in ; It tells Marta how much clay is needed to make the model. b. Double r. 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled.

c. Both the radius and the height are doubled. 2 The volume is multiplied by 2 or 4. SOLUTION: Find the volume of the original cone. Then alter the c. Double r and h. values.

volume is multiplied by 23 or 8.

a. Double h. ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic The volume is doubled. centimeters and a height of 11.5 centimeters. Find

the side length of the base. b. Double r. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

3 volume is multiplied by 2 or 8. The side length of the base is 15 cm. ANSWER: a. The volume is doubled. ANSWER: 2 b. The volume is multiplied by 2 or 4. 15 cm c. The volume is multiplied by 23 or 8. 33. The volume of a cone is 196π cubic inches and the Find each measure. Round to the nearest tenth height is 12 inches. What is the diameter? if necessary. 32. A square pyramid has a volume of 862.5 cubic SOLUTION: centimeters and a height of 11.5 centimeters. Find The volume of a circular cone is , or the side length of the base.

SOLUTION: , where B is the area of the base, h is the The volume of a pyramid is , where B is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 area of the base and h is the height of the pyramid. centimeters. Let s be the side length of the base.

The diameter is 2(7) or 14 inches.

The side length of the base is 15 cm. ANSWER: 14 in. ANSWER: 15 cm 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION: SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters. The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

So, the radius is about 3.8 millimeters.

The diameter is 2(7) or 14 inches. Use the Pythagorean Theorem to find the height of

ANSWER: the cone.

14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER:

3 So, the radius is about 3.8 millimeters. 70.2 mm

Use the Pythagorean Theorem to find the height of 35. MULTIPLE REPRESENTATIONS In this the cone. problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. So, the height of the cone is about 4.64 millimeters. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24. The volume of a circular cone is , where

r is the radius of the base and h is the height of the Sample answer: cone.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. The volumes are the same. The volume of a b. VERBAL What is true about the volumes of the pyramid equals one third times the base area times two pyramids that you drew? Explain. the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes c. ANALYTICAL Explain how multiplying the base are equal. area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. ANSWER: CCSS ARGUMENTS a. Sample answer: 36. Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

are equal. The volumes will only be equal when the radius of c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the the cone is equal to or when . volume is multiplied by 5. If both the base area and Therefore, the statement is true sometimes if the the height are multiplied by 5, the volume is multiplied base area of the cone is 3 times as great as the base by 5 · 5 or 25. area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume CCSS ARGUMENTS 36. Determine whether the is 10h cubic units. So, the cone must have a base following statement is sometimes, always, or never area of 30 square units so that its volume is

true. Justify your reasoning. The volume of a cone with radius r and height h or 10h cubic units. equals the volume of a prism with height h. SOLUTION: ANSWER: The volume of a cone with a radius r and height h is Sometimes; the statement is true if the base area of the cone is 3 times as great as the base area of the . The volume of a prism with a height of prism. For example, if the base of the prism has an h is where B is the area of the base of the area of 10 square units, then its volume is 10h cubic prism. Set the volumes equal. units. So, the cone must have a base area of 30

square units so that its volume is or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

The volumes will only be equal when the radius of the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base SOLUTION: area of the prism. For example, if the base of the The slant height is used for surface area, but the prism has an area of 10 square units, then its volume height is used for volume. For this cone, the slant is 10h cubic units. So, the cone must have a base height of 13 is provided, and we need to calculate the area of 30 square units so that its volume is height before we can calculate the volume.

or 10h cubic units.

Alexandra incorrectly used the slant height. ANSWER: Sometimes; the statement is true if the base area of ANSWER: the cone is 3 times as great as the base area of the Cornelio; Alexandra incorrectly used the slant height. prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic 38. REASONING A cone has a volume of 568 cubic units. So, the cone must have a base area of 30 centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain square units so that its volume is or 10h your reasoning. cubic units. SOLUTION: 3 37. ERROR ANALYSIS Alexandra and Cornelio are 1704 cm ; The formula for the volume of a cylinder calculating the volume of the cone below. Is either of is V= Bh, while the formula for the volume of a cone them correct? Explain your answer. is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

ANSWER: 1704 cm3; The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

OPEN ENDED 39. Give an example of a pyramid and a prism that have the same base and the same SOLUTION: volume. Explain your reasoning. The slant height is used for surface area, but the SOLUTION: height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the The formula for volume of a prism is V = Bh and the height before we can calculate the volume. formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of Alexandra incorrectly used the slant height. the pyramid must be 3 times as great as the height of the prism. ANSWER: Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. Set the base areas of the prism and pyramid, and SOLUTION: make the height of the pyramid equal to 3 times the 3 1704 cm ; The formula for the volume of a cylinder height of the prism. is V= Bh, while the formula for the volume of a cone Sample answer: is V = Bh. The volume of a cylinder is three times A square pyramid with a base area of 16 and a as much as the volume of a cone with the same height of 12, a prism with a square base area of 16 radius and height. and a height of 4.

ANSWER: ANSWER: 1704 cm3; The volume of a cylinder is three times as Sample answer: A square pyramid with a base area much as the volume of a cone with the same radius of 16 and a height of 12, a prism with a square base and height. area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same 39. OPEN ENDED Give an example of a pyramid and volume, the height of the pyramid must be 3 times as a prism that have the same base and the same great as the height of the prism. volume. Explain your reasoning. 40. WRITING IN MATH Compare and contrast SOLUTION: finding volumes of pyramids and cones with finding The formula for volume of a prism is V = Bh and the volumes of prisms and cylinders. formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, SOLUTION: then in order to have the same volume, the height of To find the volume of each solid, you must know the the pyramid must be 3 times as great as the height of area of the base and the height. The volume of a the prism. pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

ANSWER: To find the volume of each solid, you must know the area of the base and the height. The volume of a Set the base areas of the prism and pyramid, and pyramid is one third the volume of a prism that has make the height of the pyramid equal to 3 times the the same height and base area. The volume of a height of the prism. cone is one third the volume of a cylinder that has the same height and base area. Sample answer: A square pyramid with a base area of 16 and a 41. A conical sand toy has the dimensions as shown height of 12, a prism with a square base area of 16 below. How many cubic centimeters of sand will it and a height of 4. hold when it is filled to the top?

ANSWER: Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base

area of 16 and a height of 4; if a pyramid and prism A 12π have the same base, then in order to have the same B volume, the height of the pyramid must be 3 times as 15π

great as the height of the prism. C 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding D volumes of prisms and cylinders. SOLUTION: SOLUTION: Use the Pythagorean Theorem to find the radius r of To find the volume of each solid, you must know the the cone. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top? So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. A 12π B 15π

SOLUTION: Therefore, the correct choice is A. Use the Pythagorean Theorem to find the radius r of the cone. ANSWER: A

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental

Therefore, the correct choice is A. probability of the spinner landing on orange?

ANSWER: A

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, F how tall is the tent’s center pole? G SOLUTION:

The volume of a pyramid is , where B is the H area of the base and h is the height of the pyramid. J

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental So, the correct choice is F.

probability of the spinner landing on orange? ANSWER: F

44. SAT/ACT For all

A –8 F B x – 4 C G D

H E

J SOLUTION:

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} So, the correct choice is E. Number of favorable outcomes: 5 ANSWER: E Find the volume of each prism.

45. So, the correct choice is F. SOLUTION: ANSWER: The volume of a prism is , where B is the F area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 44. SAT/ACT For all inches and a width of 12 inches. The height h of the prism is 6 inches. A –8 B x – 4 C D E

3 SOLUTION: Therefore, the volume of the prism is 1008 in . ANSWER: 1008 in3

So, the correct choice is E.

ANSWER: E 46. Find the volume of each prism. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will 45. bisect the base. Use the Pythagorean Theorem to SOLUTION: find the height of the triangle. The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

3 Therefore, the volume of the prism is 1008 in . So, the height of the triangle is 12 feet. Find the area ANSWER: of the triangle. 1008 in3

46. So, the area of the base B is 60 ft2. SOLUTION: The volume of a prism is , where B is the The height h of the prism is 19 feet. area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 1140 ft

47. SOLUTION: So, the height of the triangle is 12 feet. Find the area The volume of a prism is , where B is the of the triangle. area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

So, the area of the base B is 60 ft2.

Therefore, the volume of the prism is about 426,437.6 The height h of the prism is 19 feet. 3 m . ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination Therefore, the volume of the prism is 1140 ft3. hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin ANSWER: allows the grain to be emptied more easily. Use the 3 dimensions in the diagram to find the entire surface 1140 ft area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the

47. surface area of the conical top and bottom and find SOLUTION: the surface area of the cylinder and add them. The formula for finding the surface area of a cone is The volume of a prism is , where B is the area of the base and h is the height of the prism. The , where is r is the radius and l is the slant height base of this prism is a rectangle with a length of 79.4 of the cone. meters and a width of 52.5 meters. The height of the prism is 102.3 meters. Find the slant height of the conical top.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination Find the slant height of the conical bottom. hopper cone and bin used by farmers to store grain The height of the conical bottom is 28 – (5 + 12 + 2) after harvest. The cone at the bottom of the bin or 9 ft. allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant Find the slant height of the conical top. height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

Find the slant height of the conical bottom. ANSWER: The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

Find the area of each shaded region. Polygons

in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50 The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the ANSWER: cylinder + Surface area of the conical top + surface 30.4 cm2 area of the conical bottom.

50. SOLUTION: ANSWER: Area of the shaded region = Area of the circle – Area of the hexagon

Find the area of each shaded region. Polygons in 50 - 52 are regular.

A regular hexagon has 6 sides, so the measure of the 49. SOLUTION: interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the Area of the shaded region = Area of the rectangle – length of the apothem. Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: 30.4 cm2

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

Now find the area of the hexagon. A regular hexagon has 6 sides, so the measure of the

interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

ANSWER: 2 54.4 in

51.

SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

Now find the area of the hexagon. So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three isosceles triangles with central angles of or ANSWER: 120°, so the angle in the triangle created by the 2 54.4 in height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

51. SOLUTION: The shaded area is the area of the equilateral triangle Use a trigonometric ratio to find the value of b. less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

Now use the area of the isosceles triangles to find the area of the equilateral triangle. So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region. The equilateral triangle can be divided into three

isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle. Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft

Use a trigonometric ratio to find the value of b.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral Now use the area of the isosceles triangles to find triangle plus the area of the inscribed circle. Let b the area of the equilateral triangle. represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

So, the area of the equilateral triangle is about 67.3 square feet. Divide the equilateral triangle into three isosceles Subtract to find the area of shaded region. triangles with central angles of or 120°. The

angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below. Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft

Use trigonometric ratios to find the values of b and h.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown Therefore, the area of the shaded region is about below. 2 168.2 mm .

ANSWER: 2 168.2 mm

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Find the volume of each pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

1. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches. ANSWER: 132 cm3 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length ANSWER: of 8 meters and a width of 4.5 meters. The height of 3 75 in the pyramid is 5.2 meters.

2. ANSWER: SOLUTION: 3 62.4 m The volume of a pyramid is , where B is the 4. a square pyramid with a height of 14 meters and a area of the base and h is the height of the pyramid. base with 8-meter side lengths The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 SOLUTION: centimeters. The height of the pyramid is 12 centimeters. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

ANSWER: ANSWER: 132 cm3 298.7 m3 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters Find the volume of each cone. Round to the nearest tenth. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length 5. of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters. SOLUTION:

The volume of a circular cone is , or

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the ANSWER: 3 area of the base and h is the height of the pyramid. 51.3 in The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

6. SOLUTION:

ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth. Use trigonometry to find the radius r.

The volume of a circular cone is , or 5. SOLUTION: , where B is the area of the base, h is the

The volume of a circular cone is , or height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: ANSWER: 3 The volume of a circular cone is , or 51.3 in , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

SOLUTION:

ANSWER: Use trigonometry to find the radius r. 3 28.1 mm 8. a cone with a slant height of 25 meters and a radius of 15 meters The volume of a circular cone is , or SOLUTION:

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and

a radius of 1.6 millimeters So, the height of the cone is 20 meters. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. ANSWER: 4712.4 m3

8. a cone with a slant height of 25 meters and a radius The volume of a circular cone is , where B of 15 meters is the area of the base and h is the height of the

SOLUTION: cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. So, the height of the cone is 20 meters.

10. SOLUTION:

ANSWER: The volume of a pyramid is , where B is the 3 4712.4 m area of the base and h is the height of the pyramid.

SOLUTION: ANSWER: 3 The volume of a circular cone is , where B 605 in is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

ANSWER: 3 105.8 mm

10. SOLUTION:

The volume of a pyramid is , where B is the 12. area of the base and h is the height of the pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 605 in3

ANSWER: 482.1 m3

11. SOLUTION:

The volume of a pyramid is , where B is the 13. area of the base and h is the height of the pyramid. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl ANSWER: 90° triangle. 3 105.8 mm

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

ANSWER: 482.1 m3

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square 13. feet and an altitude of 7 feet SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The apothem is .

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: 1376.7 ft3 ANSWER: 3 15. a triangular pyramid with a height of 4.8 centimeters 35.6 cm and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has SOLUTION: a volume of 144 cubic centimeters. Find the height. Find the height of the right triangle. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

The length of the other leg of the right triangle is 6 cm. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

So, the area of the base B is 24 cm2.

ANSWER: Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the Therefore, the height of the triangular pyramid is 18 triangle. cm.

ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth.

17. SOLUTION:

The volume of a circular cone is ,

The length of the other leg of the right triangle is 6 where r is the radius of the base and h is the height cm. of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

3 Therefore, the volume of the cone is about 235.6 in . ANSWER: 235.6 in3

Therefore, the height of the triangular pyramid is 18 cm. 18.

ANSWER: SOLUTION: 18 cm The volume of a circular cone is , where Find the volume of each cone. Round to the r is the radius of the base and h is the height of the nearest tenth. cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height Therefore, the volume of the cone is about 134.8 3 of the cone. cm .

Since the diameter of this cone is 10 inches, the ANSWER: radius is or 5 inches. The height of the cone is 9 134.8 cm3 inches.

19. SOLUTION:

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3

Use a trigonometric ratio to find the height h of the cone.

18. SOLUTION:

The volume of a circular cone is , where The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and r is the radius of the base and h is the height of the the height is 7.3 centimeters. cone. The radius of this cone is 8 centimeters.

Therefore, the volume of the cone is about 134.8 Therefore, the volume of the cone is about 1473.1 3 3 cm . cm .

ANSWER: ANSWER: 12-5 Volumes of Pyramids and Cones 134.8 cm3 1473.1 cm3

20. 19. SOLUTION: SOLUTION:

Use trigonometric ratios to find the height h and the radius r of the cone. Use a trigonometric ratio to find the height h of the cone.

Therefore, the volume of the cone is about 1473.1 3 3 cm . Therefore, the volume of the cone is about 2.8 ft .

ANSWER: ANSWER: 3 1473.1 cm3 2.8 ft 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

20. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

eSolutions Manual - Powered by Cognero Page 8

Use trigonometric ratios to find the height h and the radius r of the cone.

Therefore, the volume of the cone is about 1072.3 3 in . ANSWER: 1072.3 in3 The volume of a circular cone is , where 22. a right cone with a slant height of 5.6 centimeters r is the radius of the base and h is the height of the and a radius of 1 centimeter cone. SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 3 2.8 ft

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3 Therefore, the volume of the cone is about 1072.3 3 in . 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a ANSWER: paper cone that is 14 centimeters high and has a base 3 diameter of 8 centimeters? Round to the nearest 1072.3 in tenth. 22. a right cone with a slant height of 5.6 centimeters SOLUTION: and a radius of 1 centimeter The volume of a circular cone is , where SOLUTION: The cone has a radius r of 1 centimeter and a slant r is the radius of the base and h is the height of the height of 5.6 centimeters. Use the Pythagorean cone. Since the diameter of the cone is 8 Theorem to find the height h of the cone. centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

CCSS MODELING 24. The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

3 Therefore, the volume of the cone is about 5.8 cm . The volume of a pyramid is , where B is the

ANSWER: area of the base and h is the height of the pyramid. 3 5.8 cm 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION: ANSWER: 3 42,000,000 ft The volume of a circular cone is , where r is the radius of the base and h is the height of the 25. GARDENING The greenhouse is a regular cone. Since the diameter of the cone is 8 octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the centimeters, the radius is or 4 centimeters. The greenhouse? height of the cone is 14 centimeters.

SOLUTION: 3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. The volume of a pyramid is , where B is the

ANSWER: area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with 3 234.6 cm sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle 24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in below is 22.5°. the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet. ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 SOLUTION: 32.2 ft

The volume of a pyramid is , where B is the Find the volume of each solid. Round to the nearest tenth. area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°. 26.

SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet.

ANSWER: 471.2 in3

27. Therefore, the volume of the greenhouse is about 32.2 ft3. SOLUTION:

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

ANSWER: 3 3190.6 m 26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

28. SOLUTION:

ANSWER: 3 ANSWER: 471.2 in 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per 27. cubic foot. What size unit does Sam need to SOLUTION: purchase?

SOLUTION: The building can be broken down into the rectangular ANSWER: base and the pyramid ceiling. The volume of the base 3 3190.6 m is

28. The volume of the ceiling is SOLUTION:

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 7698.5 cm3 ANSWER: 13,333 BTUs 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she 30. SCIENCE Marta is studying crystals that grow on needs to determine the BTUs (British Thermal Units) rock formations.For a project, she is making a clay required to heat the building. For new construction model of a crystal with a shape that is a composite of with good insulation, there should be 2 BTUs per two congruent rectangular pyramids. The base of cubic foot. What size unit does Sam need to each pyramid will be 1 by 1.5 inches, and the total purchase? height will be 4 inches.

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

It tells Marta how much clay is needed to make the model. The volume of the ceiling is ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the The total volume is therefore 5000 + 1666.67 = cone. 3 6666.67 ft . Two BTU's are needed for every cubic a. The height is doubled. foot, so the size of the heating unit Sam should buy is b. The radius is doubled. 6666.67 × 2 = 13,333 BTUs. c. Both the radius and the height are doubled.

SOLUTION: ANSWER: Find the volume of the original cone. Then alter the 13,333 BTUs values.

Determine the volume of the model. Explain why a. Double h. knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The volume is doubled.

b. Double r.

It tells Marta how much clay is needed to make the model.

ANSWER: 2 in3; It tells Marta how much clay is needed to 2 make the model. The volume is multiplied by 2 or 4.

31. CHANGING DIMENSIONS A cone has a radius c. Double r and h. of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: 3 Find the volume of the original cone. Then alter the volume is multiplied by 2 or 8. values. ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic a. Double h. centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The volume is doubled.

b. Double r.

2 The volume is multiplied by 2 or 4.

c. Double r and h. The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? 3 volume is multiplied by 2 or 8. SOLUTION:

ANSWER: The volume of a circular cone is , or a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. , where B is the area of the base, h is the 3 c. The volume is multiplied by 2 or 8. height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 Find each measure. Round to the nearest tenth centimeters. if necessary. 32. A square pyramid has a volume of 862.5 cubic

centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The diameter is 2(7) or 14 inches.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter?

SOLUTION: The lateral area of a cone is , where r is the radius and is the slant height of the cone. The volume of a circular cone is , or Replace L with 71.6 and with 6, then solve for the radius r. , where B is the area of the base, h is the

height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

The diameter is 2(7) or 14 inches.

ANSWER: So, the height of the cone is about 4.64 millimeters. 14 in. 34. The lateral area of a cone is 71.6 square millimeters The volume of a circular cone is , where and the slant height is 6 millimeters. What is the r is the radius of the base and h is the height of the volume of the cone? cone. SOLUTION:

Therefore, the volume of the cone is about 70.2 3 mm . The lateral area of a cone is , where r is ANSWER: the radius and is the slant height of the cone. 3 Replace L with 71.6 and with 6, then solve for the 70.2 mm radius r. 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. So, the radius is about 3.8 millimeters. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the Use the Pythagorean Theorem to find the height of volume of the pyramid. the cone. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm . b. The volumes are the same. The volume of a pyramid equals one third times the base area times ANSWER: the height. So, if the base areas of two pyramids are 70.2 mm3 equal and their heights are equal, then their volumes are equal. 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION:

a. Use rectangular bases and pick values that multiply to make 24. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the Sample answer: volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

ANSWER: a. Sample answer:

The volumes will only be equal when the radius of the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base b. The volumes are the same. The volume of a area of the prism. For example, if the base of the pyramid equals one third times the base area times prism has an area of 10 square units, then its volume the height. So, if the base areas of two pyramids are is 10h cubic units. So, the cone must have a base equal and their heights are equal, then their volumes area of 30 square units so that its volume is

are equal. c. If the base area is multiplied by 5, the volume is or 10h cubic units. multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and ANSWER: the height are multiplied by 5, the volume is multiplied Sometimes; the statement is true if the base area of by 5 · 5 or 25. the cone is 3 times as great as the base area of the 36. CCSS ARGUMENTS Determine whether the prism. For example, if the base of the prism has an following statement is sometimes, always, or never area of 10 square units, then its volume is 10h cubic true. Justify your reasoning. units. So, the cone must have a base area of 30 The volume of a cone with radius r and height h square units so that its volume is or 10h equals the volume of a prism with height h. cubic units. SOLUTION: The volume of a cone with a radius r and height h is 37. ERROR ANALYSIS Alexandra and Cornelio are . The volume of a prism with a height of calculating the volume of the cone below. Is either of them correct? Explain your answer. h is where B is the area of the base of the prism. Set the volumes equal.

SOLUTION: The slant height is used for surface area, but the height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume. The volumes will only be equal when the radius of the cone is equal to or when . Alexandra incorrectly used the slant height. Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base ANSWER: area of the prism. For example, if the base of the Cornelio; Alexandra incorrectly used the slant height. prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base 38. REASONING A cone has a volume of 568 cubic area of 30 square units so that its volume is centimeters. What is the volume of a cylinder that

or 10h cubic units. has the same radius and height as the cone? Explain your reasoning. ANSWER: SOLUTION: 3 Sometimes; the statement is true if the base area of 1704 cm ; The formula for the volume of a cylinder the cone is 3 times as great as the base area of the is V= Bh, while the formula for the volume of a cone prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic is V = Bh. The volume of a cylinder is three times units. So, the cone must have a base area of 30 as much as the volume of a cone with the same radius and height. square units so that its volume is or 10h ANSWER: cubic units. 1704 cm3; The volume of a cylinder is three times as 37. ERROR ANALYSIS Alexandra and Cornelio are much as the volume of a cone with the same radius calculating the volume of the cone below. Is either of and height. them correct? Explain your answer. 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism. SOLUTION: The slant height is used for surface area, but the height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume.

Alexandra incorrectly used the slant height. Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the ANSWER: height of the prism. Cornelio; Alexandra incorrectly used the slant height. Sample answer: 38. REASONING A cone has a volume of 568 cubic A square pyramid with a base area of 16 and a centimeters. What is the volume of a cylinder that height of 12, a prism with a square base area of 16 has the same radius and height as the cone? Explain and a height of 4. your reasoning. ANSWER: SOLUTION: Sample answer: A square pyramid with a base area 3 1704 cm ; The formula for the volume of a cylinder of 16 and a height of 12, a prism with a square base is V= Bh, while the formula for the volume of a cone area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same is V = Bh. The volume of a cylinder is three times volume, the height of the pyramid must be 3 times as as much as the volume of a cone with the same great as the height of the prism. radius and height. 40. WRITING IN MATH Compare and contrast ANSWER: finding volumes of pyramids and cones with finding 1704 cm3; The volume of a cylinder is three times as volumes of prisms and cylinders. much as the volume of a cone with the same radius SOLUTION: and height. To find the volume of each solid, you must know the 39. OPEN ENDED Give an example of a pyramid and area of the base and the height. The volume of a a prism that have the same base and the same pyramid is one third the volume of a prism that has volume. Explain your reasoning. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the SOLUTION: same height and base area. The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of ANSWER: that. So, if a pyramid and prism have the same base, To find the volume of each solid, you must know the then in order to have the same volume, the height of area of the base and the height. The volume of a the pyramid must be 3 times as great as the height of pyramid is one third the volume of a prism that has the prism. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism. A 12π Sample answer: B 15π A square pyramid with a base area of 16 and a

height of 12, a prism with a square base area of 16 C and a height of 4.

ANSWER: D Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base SOLUTION: area of 16 and a height of 4; if a pyramid and prism Use the Pythagorean Theorem to find the radius r of have the same base, then in order to have the same the cone. volume, the height of the pyramid must be 3 times as great as the height of the prism.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

ANSWER: To find the volume of each solid, you must know the So, the radius of the cone is 3 centimeters. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has The volume of a circular cone is , where the same height and base area. The volume of a r is the radius of the base and h is the height of the cone is one third the volume of a cylinder that has the cone. same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

Therefore, the correct choice is A.

A 12π ANSWER: B 15π A

C 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is D 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION: SOLUTION: Use the Pythagorean Theorem to find the radius r of

the cone. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 5.5 ft So, the radius of the cone is 3 centimeters. 43. PROBABILITY A spinner has sections colored The volume of a circular cone is , where red, blue, orange, and green. The table below shows the results of several spins. What is the experimental r is the radius of the base and h is the height of the probability of the spinner landing on orange? cone.

Therefore, the correct choice is A. G

ANSWER: H A

J 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, SOLUTION: how tall is the tent’s center pole? Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} SOLUTION: Number of possible outcomes : 25 Favorable outcomes: {5 orange} The volume of a pyramid is , where B is the Number of favorable outcomes: 5 area of the base and h is the height of the pyramid.

So, the correct choice is F.

ANSWER: F ANSWER: 5.5 ft 44. SAT/ACT For all

43. PROBABILITY A spinner has sections colored A –8 red, blue, orange, and green. The table below shows B x – 4 the results of several spins. What is the experimental C probability of the spinner landing on orange? D E

SOLUTION:

So, the correct choice is E. H ANSWER: J E

SOLUTION: Find the volume of each prism. Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange}

Number of favorable outcomes: 5 45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the correct choice is F.

ANSWER: F

SAT/ACT 3 44. For all Therefore, the volume of the prism is 1008 in .

A –8 ANSWER: B x – 4 3 C 1008 in D E

SOLUTION: 46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base So, the correct choice is E. of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to ANSWER: find the height of the triangle. E Find the volume of each prism.

45.

prism is 6 inches. So, the height of the triangle is 12 feet. Find the area of the triangle.

3 Therefore, the volume of the prism is 1008 in . 2 ANSWER: So, the area of the base B is 60 ft .

1008 in3 The height h of the prism is 19 feet.

46. Therefore, the volume of the prism is 1140 ft3. SOLUTION: The volume of a prism is , where B is the ANSWER: area of the base and h is the height of the prism. The 3 1140 ft base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the height of the triangle is 12 feet. Find the area of the triangle. Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination 2 So, the area of the base B is 60 ft . hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin The height h of the prism is 19 feet. allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863.

SOLUTION: Therefore, the volume of the prism is 1140 ft3. To find the entire surface area of the bin, find the ANSWER: surface area of the conical top and bottom and find

3 the surface area of the cylinder and add them. 1140 ft The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2)

or 9 ft. Therefore, the volume of the prism is about 426,437.6 3 m . ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the The formula for finding the surface area of a cylinder nearest square foot. Refer to the photo on Page 863. is , where is r is the radius and h is the slant height of the cylinder. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. Surface area of the bin = Surface area of the The formula for finding the surface area of a cone is cylinder + Surface area of the conical top + surface , where is r is the radius and l is the slant height area of the conical bottom. of the cone.

Find the slant height of the conical top.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) SOLUTION: or 9 ft. Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: 2 30.4 cm The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

50.

SOLUTION: Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface Area of the shaded region = Area of the circle –

area of the conical bottom. Area of the hexagon

A regular hexagon has 6 sides, so the measure of the ANSWER: interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem. Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: 30.4 cm2

50. Now find the area of the hexagon. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the ANSWER: length of the apothem. 2 54.4 in

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

The equilateral triangle can be divided into three

ANSWER: Use a trigonometric ratio to find the value of b. 2 54.4 in

51. Now use the area of the isosceles triangles to find SOLUTION: the area of the equilateral triangle. The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the equilateral triangle is about 67.3 square feet.

So, the area of the circle is about 40.7 square feet. Subtract to find the area of shaded region. Next, find the area of the equilateral triangle.

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER:

2 The equilateral triangle can be divided into three 26.6 ft isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle. 52.

SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral Use a trigonometric ratio to find the value of b. triangle and h represent the radius of the inscribed circle.

Now use the area of the isosceles triangles to find the area of the equilateral triangle. Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below. So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

Use trigonometric ratios to find the values of b and h. Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft

52. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find SOLUTION: the area of the shaded region. The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral A(shaded region) = A(circumscribed circle) - A triangle plus the area of the inscribed circle. Let b (equilateral triangle) + A(inscribed circle) represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Therefore, the area of the shaded region is about 168.2 mm2.

Divide the equilateral triangle into three isosceles ANSWER: 2 triangles with central angles of or 120°. The 168.2 mm angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

Find the volume of each pyramid. ANSWER: 3 75 in

1. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 2. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the SOLUTION: pyramid is 10 inches. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

ANSWER: 3 75 in

ANSWER: 132 cm3

2. 3. a rectangular pyramid with a height of 5.2 meters

SOLUTION: and a base 8 meters by 4.5 meters SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a regular pentagon with area of the base and h is the height of the pyramid. sides of 4.4 centimeters and an apothem of 3 The base of this pyramid is a rectangle with a length centimeters. The height of the pyramid is 12 of 8 meters and a width of 4.5 meters. The height of centimeters. the pyramid is 5.2 meters.

ANSWER: 3 62.4 m ANSWER: 3 4. a square pyramid with a height of 14 meters and a 132 cm base with 8-meter side lengths 3. a rectangular pyramid with a height of 5.2 meters SOLUTION: and a base 8 meters by 4.5 meters The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. area of the base and h is the height of the pyramid.

The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 298.7 m3

ANSWER: Find the volume of each cone. Round to the 3 nearest tenth. 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION: 5. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a circular cone is , or The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth. ANSWER: 3 51.3 in

5. SOLUTION:

The volume of a circular cone is , or 6. SOLUTION: , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

Use trigonometry to find the radius r.

The volume of a circular cone is , or

ANSWER: , where B is the area of the base, h is the 3 51.3 in height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

SOLUTION: ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters

Use trigonometry to find the radius r. SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the The volume of a circular cone is , or height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the , where B is the area of the base, h is the height is 10.5 millimeters. height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

ANSWER: 3 ANSWER: 28.1 mm 168.1 cm3 8. a cone with a slant height of 25 meters and a radius 7. an oblique cone with a height of 10.5 millimeters and of 15 meters a radius of 1.6 millimeters SOLUTION: SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the

height of the cone, and r is the radius of the base. Use the Pythagorean Theorem to find the height h of The radius of this cone is 1.6 millimeters and the the cone. Then find its volume. height is 10.5 millimeters.

So, the height of the cone is 20 meters. ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

ANSWER: 4712.4 m3

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical Use the Pythagorean Theorem to find the height h of building. If the height is 100 feet and the area of the the cone. Then find its volume. base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B

is the area of the base and h is the height of the So, the height of the cone is 20 meters. cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 4712.4 m3 ANSWER: 3 9. MUSEUMS The sky dome of the National Corvette 513,333.3 ft Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the CCSS SENSE-MAKING Find the volume of base is about 15,400 square feet, find the volume of each pyramid. Round to the nearest tenth if air that the heating and cooling systems would have necessary. to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the 10. cone. For this cone, the area of the base is 15,400 square SOLUTION: feet and the height is 100 feet. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: 3 513,333.3 ft ANSWER: 3 CCSS SENSE-MAKING Find the volume of 605 in each pyramid. Round to the nearest tenth if necessary.

11. 10. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 3 105.8 mm 605 in3

12.

11. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

area of the base and h is the height of the pyramid.

ANSWER: 482.1 m3 ANSWER: 3 105.8 mm

13. 12. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 482.1 m3

13. SOLUTION: The apothem is .

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and ANSWER: hypotenuse 10.2 centimeters 3 233.8 cm SOLUTION: Find the height of the right triangle. 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 1376.7 ft3

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a

leg 8 centimeters and hypotenuse 10 centimeters has The length of the other leg of the right triangle is 6 a volume of 144 cubic centimeters. Find the height. cm. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The length of the other leg of the right triangle is 6 Therefore, the height of the triangular pyramid is 18 cm. cm.

ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for 17. the volume of a pyramid and solve for the height h. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches. Therefore, the height of the triangular pyramid is 18 cm. ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth. 3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3 17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. 18.

SOLUTION: Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 The volume of a circular cone is , where inches. r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: Therefore, the volume of the cone is about 134.8 3 3 235.6 in cm .

ANSWER: 134.8 cm3

18. SOLUTION:

The volume of a circular cone is , where 19. r is the radius of the base and h is the height of the SOLUTION: cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Use a trigonometric ratio to find the height h of the cone. Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 134.8 cm3 The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

19. SOLUTION:

Therefore, the volume of the cone is about 1473.1 3 cm .

ANSWER: 3 Use a trigonometric ratio to find the height h of the 1473.1 cm cone.

20. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Use trigonometric ratios to find the height h and the radius r of the cone.

Therefore, the volume of the cone is about 1473.1 3 cm . ANSWER: 1473.1 cm3 The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

20. SOLUTION:

3 Therefore, the volume of the cone is about 2.8 ft .

Use trigonometric ratios to find the height h and the ANSWER: radius r of the cone. 3 2.8 ft 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches

SOLUTION:

The volume of a circular cone is , where The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, r is the radius of the base and h is the height of the cone. the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 3 3 Therefore, the volume of the cone is about 2.8 ft . in .

ANSWER: ANSWER: 12-5 Volumes of Pyramids and Cones 2.8 ft3 1072.3 in3

21. an oblique cone with a diameter of 16 inches and an 22. a right cone with a slant height of 5.6 centimeters altitude of 16 inches and a radius of 1 centimeter SOLUTION: SOLUTION: The cone has a radius r of 1 centimeter and a slant The volume of a circular cone is , where height of 5.6 centimeters. Use the Pythagorean r is the radius of the base and h is the height of the Theorem to find the height h of the cone. cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 3 1072.3 in

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. 3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3

SOLUTION:

eSolutions Manual - Powered by Cognero Page 9 3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 3 3 5.8 cm Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a ANSWER: paper cone that is 14 centimeters high and has a base 234.6 cm3 diameter of 8 centimeters? Round to the nearest tenth. 24. CCSS MODELING The Pyramid Arena in SOLUTION: Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its The volume of a circular cone is , where square base is 600 feet wide. Find the volume of this pyramid. r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 SOLUTION: centimeters, the radius is or 4 centimeters. The The volume of a pyramid is , where B is the height of the cone is 14 centimeters. area of the base and h is the height of the pyramid.

3 Therefore, the paper cone will hold about 234.6 cm ANSWER: of roasted peanuts. 3 42,000,000 ft ANSWER: 234.6 cm3 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base 24. CCSS MODELING The Pyramid Arena in has side lengths of 2 feet. What is the volume of the Memphis, Tennessee, is the third largest pyramid in greenhouse? the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. SOLUTION:

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Use a trigonometric ratio to find the apothem a.

SOLUTION: The height of this pyramid is 5 feet. The volume of a pyramid is , where B is the

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft Use a trigonometric ratio to find the apothem a. Find the volume of each solid. Round to the nearest tenth.

The height of this pyramid is 5 feet. 26.

SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth. ANSWER: 471.2 in3

26. SOLUTION: Volume of the solid given = Volume of the small 27. cone + Volume of the large cone SOLUTION:

ANSWER: 3 3190.6 m ANSWER: 471.2 in3

28. SOLUTION:

27. SOLUTION:

ANSWER: 7698.5 cm3 ANSWER: HEATING 3 29. Sam is building an art studio in her 3190.6 m backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

28. SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she The volume of the ceiling is needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

SOLUTION: ANSWER: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base 13,333 BTUs is 30. SCIENCE Marta is studying crystals that grow on

rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches.

The volume of the ceiling is Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs It tells Marta how much clay is needed to make the model. SCIENCE 30. Marta is studying crystals that grow on ANSWER: rock formations.For a project, she is making a clay 3 model of a crystal with a shape that is a composite of 2 in ; It tells Marta how much clay is needed to two congruent rectangular pyramids. The base of make the model. each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches. 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Determine the volume of the model. Explain why Describe how each change affects the volume of the knowing the volume is helpful in this situation. cone. a. The height is doubled. SOLUTION: b. The radius is doubled. c. The volume of a pyramid is , where B is the Both the radius and the height are doubled. area of the base and h is the height of the pyramid. SOLUTION: Find the volume of the original cone. Then alter the values.

It tells Marta how much clay is needed to make the model.

ANSWER: a. Double h. 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. The volume is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. b. Double r. SOLUTION: Find the volume of the original cone. Then alter the values.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

a. Double h.

volume is multiplied by 23 or 8.

The volume is doubled. ANSWER: a. The volume is doubled. 2 b. Double r. b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

c. Double r and h. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

volume is multiplied by 23 or 8.

ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic The side length of the base is 15 cm. centimeters and a height of 11.5 centimeters. Find ANSWER: the side length of the base. 15 cm SOLUTION:

The volume of a pyramid is , where B is the 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? area of the base and h is the height of the pyramid. Let s be the side length of the base. SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? The diameter is 2(7) or 14 inches. SOLUTION: ANSWER: 14 in. The volume of a circular cone is , or 34. The lateral area of a cone is 71.6 square millimeters , where B is the area of the base, h is the and the slant height is 6 millimeters. What is the volume of the cone? height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 SOLUTION: centimeters.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The diameter is 2(7) or 14 inches. ANSWER: 14 in.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the So, the radius is about 3.8 millimeters. volume of the cone?

SOLUTION: Use the Pythagorean Theorem to find the height of the cone.

The lateral area of a cone is , where r is So, the height of the cone is about 4.64 millimeters. the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone. Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with So, the height of the cone is about 4.64 millimeters. different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the The volume of a circular cone is , where two pyramids that you drew? Explain. r is the radius of the base and h is the height of the c. ANALYTICAL Explain how multiplying the base cone. area and/or the height of the pyramid by 5 affects the volume of the pyramid.

SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that b. The volumes are the same. The volume of a multiply to make 24. pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes Sample answer: are equal.

ANSWER: a. Sample answer:

b. The volumes are the same. The volume of a pyramid equals one third times the base area times ANSWER: the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes a. Sample answer: are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

CCSS ARGUMENTS 36. Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volumes will only be equal when the radius of The volume of a cone with radius r and height h the cone is equal to or when . equals the volume of a prism with height h. Therefore, the statement is true sometimes if the SOLUTION: base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the The volume of a cone with a radius r and height h is prism has an area of 10 square units, then its volume . The volume of a prism with a height of is 10h cubic units. So, the cone must have a base h is where B is the area of the base of the area of 30 square units so that its volume is prism. Set the volumes equal. or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are The volumes will only be equal when the radius of calculating the volume of the cone below. Is either of them correct? Explain your answer. the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

ANSWER: SOLUTION: Sometimes; the statement is true if the base area of The slant height is used for surface area, but the the cone is 3 times as great as the base area of the height is used for volume. For this cone, the slant prism. For example, if the base of the prism has an height of 13 is provided, and we need to calculate the area of 10 square units, then its volume is 10h cubic height before we can calculate the volume. units. So, the cone must have a base area of 30

square units so that its volume is or 10h Alexandra incorrectly used the slant height. cubic units. ANSWER: 37. ERROR ANALYSIS Alexandra and Cornelio are Cornelio; Alexandra incorrectly used the slant height. calculating the volume of the cone below. Is either of them correct? Explain your answer. 38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. SOLUTION: 3 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height. SOLUTION: ANSWER: The slant height is used for surface area, but the height is used for volume. For this cone, the slant 1704 cm3; The volume of a cylinder is three times as height of 13 is provided, and we need to calculate the much as the volume of a cone with the same radius height before we can calculate the volume. and height.

39. OPEN ENDED Give an example of a pyramid and Alexandra incorrectly used the slant height. a prism that have the same base and the same volume. Explain your reasoning. ANSWER: Cornelio; Alexandra incorrectly used the slant height. SOLUTION: The formula for volume of a prism is V = Bh and the 38. REASONING A cone has a volume of 568 cubic formula for the volume of a pyramid is one-third of centimeters. What is the volume of a cylinder that that. So, if a pyramid and prism have the same base, has the same radius and height as the cone? Explain then in order to have the same volume, the height of your reasoning. the pyramid must be 3 times as great as the height of the prism. SOLUTION: 3 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same

radius and height. Set the base areas of the prism and pyramid, and ANSWER: make the height of the pyramid equal to 3 times the 3 1704 cm ; The volume of a cylinder is three times as height of the prism. much as the volume of a cone with the same radius and height. Sample answer: A square pyramid with a base area of 16 and a 39. OPEN ENDED Give an example of a pyramid and height of 12, a prism with a square base area of 16 a prism that have the same base and the same and a height of 4. volume. Explain your reasoning. ANSWER: SOLUTION: Sample answer: A square pyramid with a base area The formula for volume of a prism is V = Bh and the of 16 and a height of 12, a prism with a square base formula for the volume of a pyramid is one-third of area of 16 and a height of 4; if a pyramid and prism that. So, if a pyramid and prism have the same base, have the same base, then in order to have the same then in order to have the same volume, the height of volume, the height of the pyramid must be 3 times as the pyramid must be 3 times as great as the height of great as the height of the prism. the prism. 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a Set the base areas of the prism and pyramid, and pyramid is one third the volume of a prism that has make the height of the pyramid equal to 3 times the the same height and base area. The volume of a height of the prism. cone is one third the volume of a cylinder that has the same height and base area. Sample answer: A square pyramid with a base area of 16 and a ANSWER: height of 12, a prism with a square base area of 16 To find the volume of each solid, you must know the and a height of 4. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has ANSWER: the same height and base area. The volume of a Sample answer: A square pyramid with a base area cone is one third the volume of a cylinder that has the of 16 and a height of 12, a prism with a square base same height and base area. area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same 41. A conical sand toy has the dimensions as shown volume, the height of the pyramid must be 3 times as below. How many cubic centimeters of sand will it great as the height of the prism. hold when it is filled to the top?

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders.

SOLUTION: A 12π To find the volume of each solid, you must know the B 15π area of the base and the height. The volume of a C pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the D same height and base area. SOLUTION: ANSWER: Use the Pythagorean Theorem to find the radius r of To find the volume of each solid, you must know the the cone. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

A 12π B 15π

C So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where D r is the radius of the base and h is the height of the cone. SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

Therefore, the correct choice is A.

ANSWER: A

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION: So, the radius of the cone is 3 centimeters. The volume of a pyramid is , where B is the The volume of a circular cone is , where area of the base and h is the height of the pyramid. r is the radius of the base and h is the height of the cone.

Therefore, the correct choice is A.

ANSWER: ANSWER: A 5.5 ft

42. SHORT RESPONSE Brooke is buying a tent that 43. PROBABILITY A spinner has sections colored is in the shape of a rectangular pyramid. The base is red, blue, orange, and green. The table below shows 6 feet by 8 feet. If the tent holds 88 cubic feet of air, the results of several spins. What is the experimental how tall is the tent’s center pole? probability of the spinner landing on orange? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION: ANSWER: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} 5.5 ft Number of possible outcomes : 25 Favorable outcomes: {5 orange} 43. PROBABILITY A spinner has sections colored Number of favorable outcomes: 5 red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

F So, the correct choice is F. ANSWER: G F

H 44. SAT/ACT For all

J A –8 B x – 4 SOLUTION: C Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} D Number of possible outcomes : 25 E Favorable outcomes: {5 orange} Number of favorable outcomes: 5 SOLUTION:

So, the correct choice is E.

ANSWER: So, the correct choice is F. E ANSWER: F Find the volume of each prism.

44. SAT/ACT For all

A –8 45. B x – 4 SOLUTION: C The volume of a prism is , where B is the D area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 E inches and a width of 12 inches. The height h of the prism is 6 inches. SOLUTION:

So, the correct choice is E. 3 ANSWER: Therefore, the volume of the prism is 1008 in . E ANSWER: Find the volume of each prism. 1008 in3

45. SOLUTION: 46. The volume of a prism is , where B is the SOLUTION: area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 The volume of a prism is , where B is the inches and a width of 12 inches. The height h of the area of the base and h is the height of the prism. The prism is 6 inches. base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to

find the height of the triangle.

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 1008 in3

So, the height of the triangle is 12 feet. Find the area 46. of the triangle.

SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle. So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 1140 ft

So, the height of the triangle is 12 feet. Find the area of the triangle.

47. SOLUTION: The volume of a prism is , where B is the So, the area of the base B is 60 ft2. area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the The height h of the prism is 19 feet. prism is 102.3 meters.

3 Therefore, the volume of the prism is 1140 ft . Therefore, the volume of the prism is about 426,437.6 3 ANSWER: m . 3 1140 ft ANSWER: 3 426,437.6 m

surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. Therefore, the volume of the prism is about 426,437.6 3 Find the slant height of the conical top. m .

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) SOLUTION: or 9 ft. To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface Find the slant height of the conical bottom. area of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant 49. height of the cylinder. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) Surface area of the bin = Surface area of the = 50 cylinder + Surface area of the conical top + surface area of the conical bottom.

ANSWER: 30.4 cm2 ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular. 50. SOLUTION: 49. Area of the shaded region = Area of the circle – Area of the hexagon SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

ANSWER: 30.4 cm2

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Now find the area of the hexagon.

ANSWER: 2 54.4 in

51. SOLUTION: The shaded area is the area of the equilateral triangle Now find the area of the hexagon. less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

ANSWER: 2 54.4 in

The equilateral triangle can be divided into three 51. isosceles triangles with central angles of or SOLUTION: 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is The shaded area is the area of the equilateral triangle half the length b of the side of the equilateral triangle. less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

Use a trigonometric ratio to find the value of b. So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is So, the area of the equilateral triangle is about 67.3 half the length b of the side of the equilateral triangle. square feet.

Subtract to find the area of shaded region.

Use a trigonometric ratio to find the value of b. Therefore, the area of the shaded region is about 2 26.6 ft . ANSWER: 2 26.6 ft

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b So, the area of the equilateral triangle is about 67.3 represent the length of each side of the equilateral

square feet. triangle and h represent the radius of the inscribed circle. Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 2 26.6 ft . Divide the equilateral triangle into three isosceles ANSWER: triangles with central angles of or 120°. The 2 angle in the triangle formed by the height.h of the 26.6 ft isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b Use trigonometric ratios to find the values of b and represent the length of each side of the equilateral h. triangle and h represent the radius of the inscribed circle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The The area of the equilateral triangle will equal three angle in the triangle formed by the height.h of the times the area of any of the isosceles triangles. Find isosceles triangle is 60° and the base is half the the area of the shaded region. length of the side of the equilateral triangle as shown below. A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about Use trigonometric ratios to find the values of b and 2 h. 168.2 mm .

ANSWER: 2 168.2 mm

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

Find the volume of each pyramid.

ANSWER: 132 cm3

1. 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs area of the base and h is the height of the pyramid. of 9 inches and 5 inches and the height of the The base of this pyramid is a rectangle with a length pyramid is 10 inches. of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 3 ANSWER: 62.4 m 3 75 in 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 2. meters. The height of the pyramid is 14 meters. SOLUTION:

3 298.7 m Find the volume of each cone. Round to the nearest tenth.

ANSWER: 3 51.3 in

ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths 6. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8

meters. The height of the pyramid is 14 meters. Use trigonometry to find the radius r.

The volume of a circular cone is , or

, where B is the area of the base, h is the ANSWER: 3 height of the cone, and r is the radius of the base. 298.7 m The height of the cone is 11.5 centimeters. Find the volume of each cone. Round to the nearest tenth.

5. ANSWER: 3 SOLUTION: 168.1 cm

The volume of a circular cone is , or 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters , where B is the area of the base, h is the SOLUTION:

height of the cone, and r is the radius of the base. The volume of a circular cone is , or Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches. , where B is the area of the base, h is the

height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

ANSWER: 3 51.3 in

ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius 6. of 15 meters SOLUTION: SOLUTION:

Use trigonometry to find the radius r. Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. So, the height of the cone is 20 meters.

ANSWER: 3 168.1 cm ANSWER: 3 7. an oblique cone with a height of 10.5 millimeters and 4712.4 m a radius of 1.6 millimeters 9. MUSEUMS The sky dome of the National Corvette SOLUTION: Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the The volume of a circular cone is , or base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have , where B is the area of the base, h is the to accommodate. Round to the nearest tenth. height of the cone, and r is the radius of the base. SOLUTION: The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. The volume of a circular cone is , where B

is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters ANSWER: 3 SOLUTION: 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

10. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. So, the height of the cone is 20 meters.

ANSWER: 3 ANSWER: 605 in 4712.4 m3

ANSWER: 3 105.8 mm

ANSWER: 3 513,333.3 ft

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 482.1 m

ANSWER: 13. 605 in3 SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri 11. determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl SOLUTION: 90° triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 105.8 mm The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

ANSWER: The volume of a pyramid is , where B is the 482.1 m3 area of the base and h is the height of the pyramid.

13. SOLUTION: ANSWER: The volume of a pyramid is , where B is the 1376.7 ft3

base and h is the height of the pyramid. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and The base is a hexagon, so we need to make a right tri hypotenuse 10.2 centimeters determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl SOLUTION: 90° triangle. Find the height of the right triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the ANSWER: 3 area of the base and h is the height of the pyramid. 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. ANSWER: Use the Pythagorean Theorem to find the other leg a 1376.7 ft3 of the right triangle and then find the area of the triangle. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: Therefore, the height of the triangular pyramid is 18 cm. 3 35.6 cm ANSWER: 16. A triangular pyramid with a right triangle base with a 18 cm leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. Find the volume of each cone. Round to the nearest tenth. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the 17. triangle. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm. 3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. 18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18 cm.

ANSWER: 18 cm Therefore, the volume of the cone is about 134.8 Find the volume of each cone. Round to the 3 nearest tenth. cm . ANSWER: 134.8 cm3

17. SOLUTION:

The volume of a circular cone is , 19. where r is the radius of the base and h is the height of the cone. SOLUTION:

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: The volume of a circular cone is , where 235.6 in3 r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

18. SOLUTION:

The volume of a circular cone is , where Therefore, the volume of the cone is about 1473.1 3 r is the radius of the base and h is the height of the cm . cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. ANSWER:

1473.1 cm3

20. Therefore, the volume of the cone is about 134.8 SOLUTION: 3 cm .

ANSWER: 134.8 cm3

Use trigonometric ratios to find the height h and the radius r of the cone.

19. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Use a trigonometric ratio to find the height h of the cone.

2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

ANSWER: the radius is or 8 inches.

1473.1 cm3

20. SOLUTION: Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an 3 altitude of 16 inches Therefore, the volume of the cone is about 5.8 cm .

SOLUTION: ANSWER: The volume of a circular cone is , where 5.8 cm3

r is the radius of the base and h is the height of the 23. SNACKS Approximately how many cubic cone. Since the diameter of this cone is 16 inches, centimeters of roasted peanuts will completely fill a the radius is or 8 inches. paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 Therefore, the volume of the cone is about 1072.3 3 centimeters, the radius is or 4 centimeters. The in . height of the cone is 14 centimeters. ANSWER:

1072.3 in3

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION:

The cone has a radius r of 1 centimeter and a slant 3 height of 5.6 centimeters. Use the Pythagorean Therefore, the paper cone will hold about 234.6 cm Theorem to find the height h of the cone. of roasted peanuts. ANSWER: 234.6 cm3

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 3 Therefore, the volume of the cone is about 5.8 cm . 42,000,000 ft ANSWER: 12-5 Volumes of Pyramids and Cones 25. GARDENING The greenhouse is a regular 5.8 cm3 octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the 23. SNACKS Approximately how many cubic greenhouse? centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 3 234.6 cm Use a trigonometric ratio to find the apothem a.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the volume of the greenhouse is about ANSWER: 32.2 ft3. 3 42,000,000 ft ANSWER: 3 25. GARDENING The greenhouse is a regular 32.2 ft octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the Find the volume of each solid. Round to the eSolutionsgreenhouse?Manual - Powered by Cognero nearest tenth. Page 10

26.

SOLUTION: Volume of the solid given = Volume of the small SOLUTION: cone + Volume of the large cone

The volume of a pyramid is , where B is the

ANSWER: 471.2 in3

Use a trigonometric ratio to find the apothem a.

27. The height of this pyramid is 5 feet. SOLUTION:

ANSWER: Therefore, the volume of the greenhouse is about 3 3190.6 m 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth. 28. SOLUTION:

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

ANSWER: 471.2 in3

SOLUTION: The building can be broken down into the rectangular 27. base and the pyramid ceiling. The volume of the base is SOLUTION:

The volume of the ceiling is

ANSWER: 3 3190.6 m

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. 28.

SOLUTION: ANSWER: 13,333 BTUs

It tells Marta how much clay is needed to make the model.

c. Both the radius and the height are doubled. The volume of the ceiling is SOLUTION: Find the volume of the original cone. Then alter the values.

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. a. Double h. ANSWER: 13,333 BTUs

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. 2 The volume is multiplied by 2 or 4.

c. Double r and h.

It tells Marta how much clay is needed to make the model.

ANSWER: 3 2 in3; It tells Marta how much clay is needed to volume is multiplied by 2 or 8. make the model. ANSWER: 31. CHANGING DIMENSIONS A cone has a radius a. The volume is doubled. 2 of 4 centimeters and a height of 9 centimeters. b. The volume is multiplied by 2 or 4. Describe how each change affects the volume of the 3 cone. c. The volume is multiplied by 2 or 8. a. The height is doubled. Find each measure. Round to the nearest tenth b. The radius is doubled. if necessary. c. Both the radius and the height are doubled. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find SOLUTION: the side length of the base. Find the volume of the original cone. Then alter the values. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

a. Double h.

The volume is doubled.

b. Double r. The side length of the base is 15 cm. ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION: 2 The volume is multiplied by 2 or 4. The volume of a circular cone is , or c. Double r and h. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

volume is multiplied by 23 or 8.

ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic The diameter is 2(7) or 14 inches. centimeters and a height of 11.5 centimeters. Find the side length of the base. ANSWER: SOLUTION: 14 in.

The volume of a pyramid is , where B is the 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the area of the base and h is the height of the pyramid. volume of the cone? Let s be the side length of the base. SOLUTION:

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r. The side length of the base is 15 cm.

ANSWER: 15 cm

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

ANSWER: 14 in. Therefore, the volume of the cone is about 70.2 34. The lateral area of a cone is 71.6 square millimeters 3 and the slant height is 6 millimeters. What is the mm . volume of the cone? ANSWER: SOLUTION: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base The lateral area of a cone is , where r is area and/or the height of the pyramid by 5 affects the the radius and is the slant height of the cone. volume of the pyramid. Replace L with 71.6 and with 6, then solve for the radius r. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

MULTIPLE REPRESENTATIONS 35. In this problem, you will investigate rectangular pyramids. c. If the base area is multiplied by 5, the volume is a. GEOMETRIC Draw two pyramids with multiplied by 5. If the height is multiplied by 5, the different bases that have a height of 10 centimeters volume is multiplied by 5. If both the base area and and a base area of 24 square centimeters. the height are multiplied by 5, the volume is multiplied b. VERBAL What is true about the volumes of the by 5 · 5 or 25. two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

ANSWER: a. Sample answer:

or 10h cubic units.

ANSWER: Sometimes; the statement is true if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an b. The volumes are the same. The volume of a area of 10 square units, then its volume is 10h cubic pyramid equals one third times the base area times units. So, the cone must have a base area of 30 the height. So, if the base areas of two pyramids are square units so that its volume is or 10h equal and their heights are equal, then their volumes are equal. cubic units. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the 37. ERROR ANALYSIS Alexandra and Cornelio are volume is multiplied by 5. If both the base area and calculating the volume of the cone below. Is either of the height are multiplied by 5, the volume is multiplied them correct? Explain your answer. by 5 · 5 or 25.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of SOLUTION: h is where B is the area of the base of the The slant height is used for surface area, but the prism. Set the volumes equal. height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning.

The volumes will only be equal when the radius of SOLUTION: 3 the cone is equal to or when . 1704 cm ; The formula for the volume of a cylinder Therefore, the statement is true sometimes if the is V= Bh, while the formula for the volume of a cone base area of the cone is 3 times as great as the base is V = Bh. The volume of a cylinder is three times area of the prism. For example, if the base of the as much as the volume of a cone with the same prism has an area of 10 square units, then its volume radius and height. is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is ANSWER: 3 or 10h cubic units. 1704 cm ; The volume of a cylinder is three times as much as the volume of a cone with the same radius and height. ANSWER: Sometimes; the statement is true if the base area of 39. OPEN ENDED Give an example of a pyramid and the cone is 3 times as great as the base area of the a prism that have the same base and the same prism. For example, if the base of the prism has an volume. Explain your reasoning. area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 SOLUTION: The formula for volume of a prism is V = Bh and the square units so that its volume is or 10h formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, cubic units. then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of 37. ERROR ANALYSIS Alexandra and Cornelio are the prism. calculating the volume of the cone below. Is either of

them correct? Explain your answer.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

your reasoning. SOLUTION: SOLUTION: To find the volume of each solid, you must know the 3 area of the base and the height. The volume of a 1704 cm ; The formula for the volume of a cylinder pyramid is one third the volume of a prism that has is V= Bh, while the formula for the volume of a cone the same height and base area. The volume of a is V = Bh. The volume of a cylinder is three times cone is one third the volume of a cylinder that has the as much as the volume of a cone with the same same height and base area. radius and height. ANSWER: ANSWER: To find the volume of each solid, you must know the 1704 cm3; The volume of a cylinder is three times as area of the base and the height. The volume of a much as the volume of a cone with the same radius pyramid is one third the volume of a prism that has and height. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the 39. OPEN ENDED Give an example of a pyramid and same height and base area. a prism that have the same base and the same volume. Explain your reasoning. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it SOLUTION: hold when it is filled to the top? The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of A 12π the prism. B 15π

SOLUTION: Set the base areas of the prism and pyramid, and Use the Pythagorean Theorem to find the radius r of make the height of the pyramid equal to 3 times the the cone. height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding So, the radius of the cone is 3 centimeters.

volumes of prisms and cylinders. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the To find the volume of each solid, you must know the cone. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

The volume of a pyramid is , where B is the

A 12π area of the base and h is the height of the pyramid.

B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

F So, the radius of the cone is 3 centimeters.

G The volume of a circular cone is , where

r is the radius of the base and h is the height of the H cone.

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Therefore, the correct choice is A. Number of favorable outcomes: 5

ANSWER: A

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? So, the correct choice is F. SOLUTION: ANSWER: The volume of a pyramid is , where B is the F area of the base and h is the height of the pyramid.

44. SAT/ACT For all

A –8 B x – 4 C D E

SOLUTION: ANSWER: 5.5 ft

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 1008 in3

So, the correct choice is F.

ANSWER: F 46. 44. SAT/ACT For all SOLUTION: The volume of a prism is , where B is the A –8 area of the base and h is the height of the prism. The B x – 4 base of this prism is an isosceles triangle with a base C of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to D find the height of the triangle. E

SOLUTION:

So, the correct choice is E.

ANSWER: E

Find the volume of each prism. So, the height of the triangle is 12 feet. Find the area of the triangle.

Therefore, the volume of the prism is 1140 ft3. 3 Therefore, the volume of the prism is 1008 in . ANSWER: ANSWER: 3 1140 ft 1008 in3

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin So, the height of the triangle is 12 feet. Find the area allows the grain to be emptied more easily. Use the of the triangle. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find So, the area of the base B is 60 ft2. the surface area of the cylinder and add them. The formula for finding the surface area of a cone is The height h of the prism is 19 feet. , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 1140 ft

Find the slant height of the conical bottom. 47. The height of the conical bottom is 28 – (5 + 12 + 2) SOLUTION: or 9 ft.

Therefore, the volume of the prism is about 426,437.6 m3. The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant ANSWER: height of the cylinder. 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain Surface area of the bin = Surface area of the after harvest. The cone at the bottom of the bin cylinder + Surface area of the conical top + surface allows the grain to be emptied more easily. Use the area of the conical bottom. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the ANSWER: surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is Find the area of each shaded region. Polygons , where is r is the radius and l is the slant height in 50 - 52 are regular. of the cone.

Find the slant height of the conical top. 49.

SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) ANSWER: or 9 ft. 30.4 cm2

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Now find the area of the hexagon.

ANSWER: 30.4 cm2

50. SOLUTION: ANSWER: 2 Area of the shaded region = Area of the circle – 54.4 in Area of the hexagon

51. SOLUTION: A regular hexagon has 6 sides, so the measure of the The shaded area is the area of the equilateral triangle interior angle is . The apothem bisects the less the area of the inscribed circle. angle, so we use 30° when using trig to find the length of the apothem. Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three

Now find the area of the hexagon. Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

ANSWER: 2 54.4 in

So, the area of the equilateral triangle is about 67.3 square feet.

51. Subtract to find the area of shaded region. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral The equilateral triangle can be divided into three triangle and h represent the radius of the inscribed circle. isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The

angle in the triangle formed by the height.h of the Use a trigonometric ratio to find the value of b. isosceles triangle is 60° and the base is half the

length of the side of the equilateral triangle as shown below.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and h.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region. Therefore, the area of the shaded region is about

2 26.6 ft . A(shaded region) = A(circumscribed circle) - A

ANSWER: (equilateral triangle) + A(inscribed circle) 2 26.6 ft

Therefore, the area of the shaded region is about 52. 168.2 mm2. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral ANSWER: 2 triangle plus the area of the inscribed circle. Let b 168.2 mm represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

Find the volume of each pyramid.

1. ANSWER: SOLUTION: 132 cm3 The volume of a pyramid is , where B is the 3. a rectangular pyramid with a height of 5.2 meters area of the base and h is the height of the pyramid. and a base 8 meters by 4.5 meters The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the SOLUTION: pyramid is 10 inches. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 3 75 in ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION: 2. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. area of the base and h is the height of the pyramid.

The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth.

ANSWER: 132 cm3 5. 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

SOLUTION: The volume of a circular cone is , or The volume of a pyramid is , where B is the , where B is the area of the base, h is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length height of the cone, and r is the radius of the base. of 8 meters and a width of 4.5 meters. The height of Since the diameter of this cone is 7 inches, the radius the pyramid is 5.2 meters. is or 3.5 inches. The height of the cone is 4 inches.

ANSWER: ANSWER: 3 3 62.4 m 51.3 in 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the 6. area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 SOLUTION: meters. The height of the pyramid is 14 meters.

Use trigonometry to find the radius r.

ANSWER: 298.7 m3 The volume of a circular cone is , or

Find the volume of each cone. Round to the , where B is the area of the base, h is the nearest tenth. height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

5. SOLUTION:

The volume of a circular cone is , or ANSWER: 3 , where B is the area of the base, h is the 168.1 cm height of the cone, and r is the radius of the base. 7. an oblique cone with a height of 10.5 millimeters and Since the diameter of this cone is 7 inches, the radius a radius of 1.6 millimeters is or 3.5 inches. The height of the cone is 4 inches. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. ANSWER: 3 51.3 in

ANSWER: 6. 3 28.1 mm SOLUTION: 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

Use trigonometry to find the radius r.

The volume of a circular cone is , or Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

, where B is the area of the base, h is the

height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

So, the height of the cone is 20 meters.

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: ANSWER: 4712.4 m3 The volume of a circular cone is , or 9. MUSEUMS The sky dome of the National Corvette , where B is the area of the base, h is the Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the height of the cone, and r is the radius of the base. base is about 15,400 square feet, find the volume of The radius of this cone is 1.6 millimeters and the air that the heating and cooling systems would have height is 10.5 millimeters. to accommodate. Round to the nearest tenth.

SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square ANSWER: feet and the height is 100 feet. 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of Use the Pythagorean Theorem to find the height h of each pyramid. Round to the nearest tenth if the cone. Then find its volume. necessary.

10. SOLUTION: So, the height of the cone is 20 meters. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 4712.4 m3 ANSWER: 9. MUSEUMS The sky dome of the National Corvette 3 Museum in Bowling Green, Kentucky, is a conical 605 in building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION: 11. The volume of a circular cone is , where B SOLUTION: is the area of the base and h is the height of the cone. The volume of a pyramid is , where B is the For this cone, the area of the base is 15,400 square feet and the height is 100 feet. area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 3 3 513,333.3 ft 105.8 mm

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

12. SOLUTION:

The volume of a pyramid is , where B is the 10. area of the base and h is the height of the pyramid. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 482.1 m3

ANSWER: 605 in3

13. SOLUTION:

The volume of a pyramid is , where B is the 11. base and h is the height of the pyramid. SOLUTION: The base is a hexagon, so we need to make a right tri The volume of a pyramid is , where B is the determine the apothem. The interior angles of the he area of the base and h is the height of the pyramid. 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 3 105.8 mm

The apothem is . 12.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 233.8 cm

ANSWER: 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet 482.1 m3 SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid. ANSWER: The base is a hexagon, so we need to make a right tri 3 1376.7 ft determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 15. a triangular pyramid with a height of 4.8 centimeters 90° triangle. and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The apothem is .

ANSWER: 3 233.8 cm The volume of a pyramid is , where B is the 14. a pentagonal pyramid with a base area of 590 square area of the base and h is the height of the pyramid. feet and an altitude of 7 feet

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. ANSWER: 1376.7 ft3 SOLUTION: The base of the pyramid is a right triangle with a leg 15. a triangular pyramid with a height of 4.8 centimeters of 8 centimeters and a hypotenuse of 10 centimeters. and a right triangle base with a leg 5 centimeters and Use the Pythagorean Theorem to find the other leg a hypotenuse 10.2 centimeters of the right triangle and then find the area of the triangle. SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

The volume of a pyramid is , where B is the So, the area of the base B is 24 cm2. area of the base and h is the height of the pyramid. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a Therefore, the height of the triangular pyramid is 18 leg 8 centimeters and hypotenuse 10 centimeters has cm. a volume of 144 cubic centimeters. Find the height. ANSWER: SOLUTION: 18 cm The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Find the volume of each cone. Round to the Use the Pythagorean Theorem to find the other leg a nearest tenth. of the right triangle and then find the area of the triangle.

17. SOLUTION:

The volume of a circular cone is ,

where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 2 So, the area of the base B is 24 cm . 235.6 in3

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and Therefore, the height of the triangular pyramid is 18 the height is 7.3 centimeters. cm. ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth.

Therefore, the volume of the cone is about 134.8 3 cm .

17. ANSWER: SOLUTION: 134.8 cm3 The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the 19. radius is or 5 inches. The height of the cone is 9 SOLUTION: inches.

Use a trigonometric ratio to find the height h of the 3 Therefore, the volume of the cone is about 235.6 in . cone.

ANSWER: 235.6 in3

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. 18. SOLUTION:

ANSWER: 1473.1 cm3

Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 20. 134.8 cm3 SOLUTION:

19. Use trigonometric ratios to find the height h and the SOLUTION: radius r of the cone.

Use a trigonometric ratio to find the height h of the The volume of a circular cone is , where cone. r is the radius of the base and h is the height of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches Therefore, the volume of the cone is about 1473.1 3 cm . SOLUTION:

ANSWER: The volume of a circular cone is , where 1473.1 cm3 r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

20. SOLUTION:

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: Use trigonometric ratios to find the height h and the 3 1072.3 in radius r of the cone. 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION:

The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where 3 Therefore, the volume of the cone is about 5.8 cm . r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, ANSWER: the radius is or 8 inches. 5.8 cm3

SOLUTION: Therefore, the volume of the cone is about 1072.3 3 The volume of a circular cone is , where in . r is the radius of the base and h is the height of the ANSWER: cone. Since the diameter of the cone is 8 3 1072.3 in centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters. 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter

SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: ANSWER: 5.8 cm3 3 42,000,000 ft 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a 25. GARDENING The greenhouse is a regular paper cone that is 14 centimeters high and has a base octagonal pyramid with a height of 5 feet. The base diameter of 8 centimeters? Round to the nearest has side lengths of 2 feet. What is the volume of the tenth. greenhouse? SOLUTION:

3 below is 22.5°. Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in

the world. It is approximately 350 feet tall, and its Use a trigonometric ratio to find the apothem a. square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The height of this pyramid is 5 feet.

ANSWER: 3 42,000,000 ft 25. GARDENING The greenhouse is a regular Therefore, the volume of the greenhouse is about 3 octagonal pyramid with a height of 5 feet. The base 32.2 ft . has side lengths of 2 feet. What is the volume of the greenhouse? ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

SOLUTION: 26. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. Volume of the solid given = Volume of the small The base of the pyramid is a regular octagon with cone + Volume of the large cone sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Use a trigonometric ratio to find the apothem a. ANSWER: 471.2 in3

The height of this pyramid is 5 feet.

27. SOLUTION:

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: ANSWER: 12-5 Volumes of Pyramids and Cones 3 3 32.2 ft 3190.6 m Find the volume of each solid. Round to the nearest tenth.

28. 26. SOLUTION: SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction ANSWER: with good insulation, there should be 2 BTUs per 3 471.2 in cubic foot. What size unit does Sam need to purchase?

27.

SOLUTION: SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

ANSWER: The volume of the ceiling is 3 3190.6 m

eSolutions Manual - Powered by Cognero Page 11 28. The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic SOLUTION: foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on

rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of ANSWER: each pyramid will be 1 by 1.5 inches, and the total 3 7698.5 cm height will be 4 inches.

29. HEATING Sam is building an art studio in her Determine the volume of the model. Explain why backyard. To buy a heating unit for the space, she knowing the volume is helpful in this situation. needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction SOLUTION: with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to The volume of a pyramid is , where B is the purchase? area of the base and h is the height of the pyramid.

SOLUTION: It tells Marta how much clay is needed to make the The building can be broken down into the rectangular model. base and the pyramid ceiling. The volume of the base is ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the The volume of the ceiling is cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values. The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER:

13,333 BTUs a. Double h. 30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches.

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. The volume is doubled.

SOLUTION: b. Double r. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

2 The volume is multiplied by 2 or 4.

c. It tells Marta how much clay is needed to make the Double r and h. model.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. volume is multiplied by 23 or 8. Describe how each change affects the volume of the cone. ANSWER: a. The height is doubled. a. The volume is doubled. 2 b. The radius is doubled. b. The volume is multiplied by 2 or 4. c. Both the radius and the height are doubled. c. The volume is multiplied by 23 or 8. SOLUTION: Find each measure. Round to the nearest tenth Find the volume of the original cone. Then alter the if necessary.

values. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base. a. Double h.

The volume is doubled.

b. Double r.

The side length of the base is 15 cm.

ANSWER: 15 cm

2 The volume is multiplied by 2 or 4. 33. The volume of a cone is 196π cubic inches and the

height is 12 inches. What is the diameter? c. Double r and h. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 3 volume is multiplied by 2 or 8. centimeters.

ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION: The diameter is 2(7) or 14 inches. The volume of a pyramid is , where B is the ANSWER: area of the base and h is the height of the pyramid. 14 in. Let s be the side length of the base. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The side length of the base is 15 cm. The lateral area of a cone is , where r is the radius and is the slant height of the cone. ANSWER: Replace L with 71.6 and with 6, then solve for the 15 cm radius r.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or So, the radius is about 3.8 millimeters. , where B is the area of the base, h is the Use the Pythagorean Theorem to find the height of height of the cone, and r is the radius of the base. the cone. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. The diameter is 2(7) or 14 inches.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone?

SOLUTION: Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters The lateral area of a cone is , where r is and a base area of 24 square centimeters. the radius and is the slant height of the cone. b. VERBAL What is true about the volumes of the Replace L with 71.6 and with 6, then solve for the two pyramids that you drew? Explain. radius r. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

So, the radius is about 3.8 millimeters. Sample answer:

Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER:

70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters

and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the c. If the base area is multiplied by 5, the volume is two pyramids that you drew? Explain. multiplied by 5. If the height is multiplied by 5, the c. ANALYTICAL Explain how multiplying the base volume is multiplied by 5. If both the base area and area and/or the height of the pyramid by 5 affects the the height are multiplied by 5, the volume is multiplied volume of the pyramid. by 5 · 5 or 25. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

ANSWER: a. Sample answer:

or 10h cubic units.

b. The volumes are the same. The volume of a ANSWER: pyramid equals one third times the base area times Sometimes; the statement is true if the base area of the height. So, if the base areas of two pyramids are the cone is 3 times as great as the base area of the equal and their heights are equal, then their volumes prism. For example, if the base of the prism has an are equal. area of 10 square units, then its volume is 10h cubic c. If the base area is multiplied by 5, the volume is units. So, the cone must have a base area of 30 multiplied by 5. If the height is multiplied by 5, the square units so that its volume is or 10h volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied cubic units. by 5 · 5 or 25. 37. ERROR ANALYSIS Alexandra and Cornelio are 36. CCSS ARGUMENTS Determine whether the calculating the volume of the cone below. Is either of following statement is sometimes, always, or never them correct? Explain your answer. true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic The volumes will only be equal when the radius of centimeters. What is the volume of a cylinder that the cone is equal to or when . has the same radius and height as the cone? Explain your reasoning. Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base SOLUTION: area of the prism. For example, if the base of the 3 prism has an area of 10 square units, then its volume 1704 cm ; The formula for the volume of a cylinder is 10h cubic units. So, the cone must have a base is V= Bh, while the formula for the volume of a cone area of 30 square units so that its volume is is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same or 10h cubic units. radius and height.

ANSWER: ANSWER: Sometimes; the statement is true if the base area of 1704 cm3; The volume of a cylinder is three times as the cone is 3 times as great as the base area of the much as the volume of a cone with the same radius prism. For example, if the base of the prism has an and height. area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same square units so that its volume is or 10h volume. Explain your reasoning. cubic units. SOLUTION: 37. ERROR ANALYSIS Alexandra and Cornelio are The formula for volume of a prism is V = Bh and the calculating the volume of the cone below. Is either of formula for the volume of a pyramid is one-third of them correct? Explain your answer. that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and SOLUTION: make the height of the pyramid equal to 3 times the The slant height is used for surface area, but the height of the prism. height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the Sample answer: height before we can calculate the volume. A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4. Alexandra incorrectly used the slant height. ANSWER: ANSWER: Sample answer: A square pyramid with a base area Cornelio; Alexandra incorrectly used the slant height. of 16 and a height of 12, a prism with a square base area of 16 and a height of 4; if a pyramid and prism 38. REASONING A cone has a volume of 568 cubic have the same base, then in order to have the same centimeters. What is the volume of a cylinder that volume, the height of the pyramid must be 3 times as has the same radius and height as the cone? Explain great as the height of the prism. your reasoning. 40. WRITING IN MATH Compare and contrast SOLUTION: finding volumes of pyramids and cones with finding 3 1704 cm ; The formula for the volume of a cylinder volumes of prisms and cylinders. is V= Bh, while the formula for the volume of a cone SOLUTION: is V = Bh. The volume of a cylinder is three times To find the volume of each solid, you must know the as much as the volume of a cone with the same area of the base and the height. The volume of a radius and height. pyramid is one third the volume of a prism that has the same height and base area. The volume of a ANSWER: cone is one third the volume of a cylinder that has the 1704 cm3; The volume of a cylinder is three times as same height and base area. much as the volume of a cone with the same radius and height. ANSWER: To find the volume of each solid, you must know the 39. OPEN ENDED Give an example of a pyramid and area of the base and the height. The volume of a a prism that have the same base and the same pyramid is one third the volume of a prism that has volume. Explain your reasoning. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the SOLUTION: same height and base area. The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of 41. A conical sand toy has the dimensions as shown that. So, if a pyramid and prism have the same base, below. How many cubic centimeters of sand will it then in order to have the same volume, the height of hold when it is filled to the top? the pyramid must be 3 times as great as the height of the prism.

A 12π B 15π

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the D height of the prism. SOLUTION: Sample answer: Use the Pythagorean Theorem to find the radius r of A square pyramid with a base area of 16 and a the cone. height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the So, the radius of the cone is 3 centimeters. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has The volume of a circular cone is , where the same height and base area. The volume of a r is the radius of the base and h is the height of the cone is one third the volume of a cylinder that has the cone. same height and base area. ANSWER: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area. Therefore, the correct choice is A.

how tall is the tent’s center pole? A 12π SOLUTION: B 15π The volume of a pyramid is , where B is the C area of the base and h is the height of the pyramid.

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where

F r is the radius of the base and h is the height of the cone. G

SOLUTION: Therefore, the correct choice is A. Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} ANSWER: Number of possible outcomes : 25 A Favorable outcomes: {5 orange} Number of favorable outcomes: 5 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. So, the correct choice is F.

ANSWER: F

44. SAT/ACT For all

A –8 B x – 4 C D ANSWER: 5.5 ft E

43. PROBABILITY A spinner has sections colored SOLUTION: red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the correct choice is E.

ANSWER: E F Find the volume of each prism.

H 45. J SOLUTION: SOLUTION: The volume of a prism is , where B is the Possible outcomes: {6 red, 4 blue, 5 orange, 10 area of the base and h is the height of the prism. The green} base of this prism is a rectangle with a length of 14 Number of possible outcomes : 25 inches and a width of 12 inches. The height h of the Favorable outcomes: {5 orange} prism is 6 inches. Number of favorable outcomes: 5

3 Therefore, the volume of the prism is 1008 in .

So, the correct choice is F. ANSWER: 3 ANSWER: 1008 in F

44. SAT/ACT For all

A –8 46. B x – 4 C SOLUTION: The volume of a prism is , where B is the D area of the base and h is the height of the prism. The E base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to SOLUTION: find the height of the triangle.

So, the correct choice is E.

ANSWER:

Find the volume of each prism.

45. So, the height of the triangle is 12 feet. Find the area of the triangle. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 3 1008 in3 Therefore, the volume of the prism is 1140 ft . ANSWER: 3 1140 ft

46. SOLUTION: The volume of a prism is , where B is the 47. area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base SOLUTION: of 10 feet and two legs of 13 feet. The height h will The volume of a prism is , where B is the bisect the base. Use the Pythagorean Theorem to area of the base and h is the height of the prism. The

find the height of the triangle. base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m So, the height of the triangle is 12 feet. Find the area of the triangle. 48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. So, the area of the base B is 60 ft2. SOLUTION: The height h of the prism is 19 feet. To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Therefore, the volume of the prism is 1140 ft3. Find the slant height of the conical top.

ANSWER: 3 1140 ft

47. SOLUTION:

The volume of a prism is , where B is the Find the slant height of the conical bottom. area of the base and h is the height of the prism. The The height of the conical bottom is 28 – (5 + 12 + 2) base of this prism is a rectangle with a length of 79.4 or 9 ft. meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m The formula for finding the surface area of a cylinder 48. FARMING The picture shows a combination is , where is r is the radius and h is the slant hopper cone and bin used by farmers to store grain height of the cylinder. after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write Surface area of the bin = Surface area of the the exact answer and the answer rounded to the cylinder + Surface area of the conical top + surface nearest square foot. area of the conical bottom. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height ANSWER: of the cone.

Find the slant height of the conical top. Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

ANSWER: 30.4 cm2

50. SOLUTION: Area of the shaded region = Area of the circle – The formula for finding the surface area of a cylinder Area of the hexagon is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the A regular hexagon has 6 sides, so the measure of the cylinder + Surface area of the conical top + surface area of the conical bottom. interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION:

Area of the shaded region = Area of the rectangle –

Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: Now find the area of the hexagon.

2 30.4 cm

50.

SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon ANSWER: 2 54.4 in

A regular hexagon has 6 sides, so the measure of the

interior angle is . The apothem bisects the 51. angle, so we use 30° when using trig to find the length of the apothem. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b.

ANSWER: Now use the area of the isosceles triangles to find

2 the area of the equilateral triangle. 54.4 in

51. SOLUTION: So, the area of the equilateral triangle is about 67.3 square feet. The shaded area is the area of the equilateral triangle less the area of the inscribed circle. Subtract to find the area of shaded region. Find the area of the circle with a radius of 3.6 feet.

Therefore, the area of the shaded region is about 2 So, the area of the circle is about 40.7 square feet. 26.6 ft .

ANSWER: Next, find the area of the equilateral triangle. 2 26.6 ft

52. SOLUTION: The equilateral triangle can be divided into three The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral isosceles triangles with central angles of or triangle plus the area of the inscribed circle. Let b 120°, so the angle in the triangle created by the represent the length of each side of the equilateral height of the isosceles triangle is 60° and the base is triangle and h represent the radius of the inscribed half the length b of the side of the equilateral triangle. circle.

Use a trigonometric ratio to find the value of b. Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below. Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and So, the area of the equilateral triangle is about 67.3 h. square feet.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about

2 26.6 ft . The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find ANSWER: the area of the shaded region. 2 26.6 ft A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral Therefore, the area of the shaded region is about triangle plus the area of the inscribed circle. Let b 168.2 mm2. represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle. ANSWER: 2 168.2 mm

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with Find the volume of each pyramid. sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

1. SOLUTION:

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

2. ANSWER: 3 SOLUTION: 62.4 m

The volume of a pyramid is , where B is the 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with SOLUTION: sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 The volume of a pyramid is , where B is the centimeters. area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

ANSWER: ANSWER: 3 132 cm3 298.7 m

3. a rectangular pyramid with a height of 5.2 meters Find the volume of each cone. Round to the and a base 8 meters by 4.5 meters nearest tenth. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 5. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of SOLUTION: the pyramid is 5.2 meters. The volume of a circular cone is , or

ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION: ANSWER: The volume of a pyramid is , where B is the 3 51.3 in area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

6. SOLUTION:

ANSWER: 298.7 m3

Find the volume of each cone. Round to the Use trigonometry to find the radius r. nearest tenth.

The volume of a circular cone is , or 5. , where B is the area of the base, h is the SOLUTION: height of the cone, and r is the radius of the base. The volume of a circular cone is , or The height of the cone is 11.5 centimeters.

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

ANSWER: The volume of a circular cone is , or 3 51.3 in , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

6. SOLUTION:

ANSWER: 3 Use trigonometry to find the radius r. 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and So, the height of the cone is 20 meters. a radius of 1.6 millimeters

SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the ANSWER: height is 10.5 millimeters. 3 4712.4 m

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. ANSWER: SOLUTION: 3 28.1 mm The volume of a circular cone is , where B 8. a cone with a slant height of 25 meters and a radius is the area of the base and h is the height of the of 15 meters cone. SOLUTION: For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

So, the height of the cone is 20 meters.

10. SOLUTION:

The volume of a pyramid is , where B is the ANSWER:

3 area of the base and h is the height of the pyramid. 4712.4 m

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. ANSWER: SOLUTION: 605 in3 The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

11.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. ANSWER: 3 105.8 mm

10. SOLUTION: 12. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: 605 in3 ANSWER: 482.1 m3

11. SOLUTION: 13. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle. ANSWER: 3 105.8 mm

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The apothem is .

ANSWER: 482.1 m3

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet 13. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The apothem is .

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

ANSWER: 3 ANSWER: 1376.7 ft 3 35.6 cm 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and 16. A triangular pyramid with a right triangle base with a hypotenuse 10.2 centimeters leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: Find the height of the right triangle. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

The length of the other leg of the right triangle is 6 cm. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for ANSWER: the volume of a pyramid and solve for the height h. 3 35.6 cm 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a Therefore, the height of the triangular pyramid is 18 of the right triangle and then find the area of the cm. triangle. ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height The length of the other leg of the right triangle is 6 of the cone. cm. Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for

the volume of a pyramid and solve for the height h. 3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3

Therefore, the height of the triangular pyramid is 18 18. cm. SOLUTION: ANSWER: The volume of a circular cone is , where 18 cm r is the radius of the base and h is the height of the Find the volume of each cone. Round to the cone. The radius of this cone is 4.2 centimeters and nearest tenth. the height is 7.3 centimeters.

17. SOLUTION:

The volume of a circular cone is , Therefore, the volume of the cone is about 134.8 3 where r is the radius of the base and h is the height cm . of the cone. ANSWER: Since the diameter of this cone is 10 inches, the 134.8 cm3 radius is or 5 inches. The height of the cone is 9 inches.

19. SOLUTION:

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3

Use a trigonometric ratio to find the height h of the cone.

18. SOLUTION:

The volume of a circular cone is , where The volume of a circular cone is , where r is the radius of the base and h is the height of the r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and cone. The radius of this cone is 8 centimeters. the height is 7.3 centimeters.

Therefore, the volume of the cone is about 1473.1 3 Therefore, the volume of the cone is about 134.8 cm . 3 cm . ANSWER: ANSWER: 1473.1 cm3 134.8 cm3

20. 19. SOLUTION: SOLUTION:

Use trigonometric ratios to find the height h and the radius r of the cone.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 1473.1 Therefore, the volume of the cone is about 2.8 ft . 3 cm . ANSWER: ANSWER: 2.8 ft3 1473.1 cm3 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where 20. r is the radius of the base and h is the height of the SOLUTION: cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Use trigonometric ratios to find the height h and the radius r of the cone.

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

The volume of a circular cone is , where 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter r is the radius of the base and h is the height of the cone. SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 3 2.8 ft 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3

Therefore, the volume of the cone is about 1072.3 23. SNACKS Approximately how many cubic 3 in . centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base ANSWER: diameter of 8 centimeters? Round to the nearest tenth. 1072.3 in3 SOLUTION: 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the The cone has a radius r of 1 centimeter and a slant cone. Since the diameter of the cone is 8 height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. ANSWER: 234.6 cm3

The volume of a pyramid is , where B is the 3 Therefore, the volume of the cone is about 5.8 cm . area of the base and h is the height of the pyramid. ANSWER: 3 5.8 cm

The volume of a circular cone is , where 25. GARDENING The greenhouse is a regular r is the radius of the base and h is the height of the octagonal pyramid with a height of 5 feet. The base cone. Since the diameter of the cone is 8 has side lengths of 2 feet. What is the volume of the greenhouse? centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

SOLUTION:

3 Therefore, the paper cone will hold about 234.6 cm The volume of a pyramid is , where B is the of roasted peanuts. area of the base and h is the height of the pyramid. ANSWER: The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is 234.6 cm3 or 45°, so the angle formed in the triangle 24. CCSS MODELING The Pyramid Arena in below is 22.5°. Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet.

ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft SOLUTION: Find the volume of each solid. Round to the The volume of a pyramid is , where B is the nearest tenth. area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle

below is 22.5°. 26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet. ANSWER: 471.2 in3

27.

Therefore, the volume of the greenhouse is about SOLUTION: 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

ANSWER: 3 3190.6 m 26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

28. SOLUTION:

ANSWER: ANSWER: 3 471.2 in 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to 27. purchase? SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base ANSWER: is 3 3190.6 m

The volume of the ceiling is 28. SOLUTION:

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is

6666.67 × 2 = 13,333 BTUs.

ANSWER: 12-5 Volumes of Pyramids and Cones ANSWER: 7698.5 cm3 13,333 BTUs

29. HEATING Sam is building an art studio in her 30. SCIENCE Marta is studying crystals that grow on backyard. To buy a heating unit for the space, she rock formations.For a project, she is making a clay needs to determine the BTUs (British Thermal Units) model of a crystal with a shape that is a composite of required to heat the building. For new construction two congruent rectangular pyramids. The base of with good insulation, there should be 2 BTUs per each pyramid will be 1 by 1.5 inches, and the total cubic foot. What size unit does Sam need to height will be 4 inches. purchase? Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

It tells Marta how much clay is needed to make the model.

The volume of the ceiling is ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. The total volume is therefore 5000 + 1666.67 = a. The height is doubled. 3 6666.67 ft . Two BTU's are needed for every cubic b. The radius is doubled. foot, so the size of the heating unit Sam should buy is c. Both the radius and the height are doubled. 6666.67 × 2 = 13,333 BTUs. SOLUTION: Find the volume of the original cone. Then alter the ANSWER: values. 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches. a. Double h. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION: eSolutions Manual - Powered by Cognero Page 12 The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The volume is doubled.

b. Double r.

It tells Marta how much clay is needed to make the model.

ANSWER: 2 2 in3; It tells Marta how much clay is needed to The volume is multiplied by 2 or 4. make the model. c. Double r and h. 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. 3 SOLUTION: volume is multiplied by 2 or 8. Find the volume of the original cone. Then alter the ANSWER: values. a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find a. Double h. the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The volume is doubled.

b. Double r.

2 The volume is multiplied by 2 or 4.

c. Double r and h. The side length of the base is 15 cm. ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION: volume is multiplied by 23 or 8. The volume of a circular cone is , or ANSWER: a. The volume is doubled. , where B is the area of the base, h is the 2 b. The volume is multiplied by 2 or 4. height of the cone, and r is the radius of the base. 3 c. The volume is multiplied by 2 or 8. Since the diameter is 8 centimeters, the radius is 4 centimeters. Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The diameter is 2(7) or 14 inches.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? The lateral area of a cone is , where r is SOLUTION: the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the The volume of a circular cone is , or radius r.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

The diameter is 2(7) or 14 inches. So, the height of the cone is about 4.64 millimeters. ANSWER: 14 in. The volume of a circular cone is , where 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the r is the radius of the base and h is the height of the

volume of the cone? cone.

SOLUTION:

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: The lateral area of a cone is , where r is 3 the radius and is the slant height of the cone. 70.2 mm Replace L with 71.6 and with 6, then solve for the radius r. 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base So, the radius is about 3.8 millimeters. area and/or the height of the pyramid by 5 affects the volume of the pyramid. Use the Pythagorean Theorem to find the height of the cone. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 b. The volumes are the same. The volume of a 3 mm . pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are ANSWER: equal and their heights are equal, then their volumes 70.2 mm3 are equal.

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid.

SOLUTION: a. Use rectangular bases and pick values that c. If the base area is multiplied by 5, the volume is multiply to make 24. multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and Sample answer: the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

ANSWER: a. Sample answer:

by 5 · 5 or 25.

. The volume of a prism with a height of ANSWER: h is where B is the area of the base of the a. Sample answer: prism. Set the volumes equal.

The volumes will only be equal when the radius of the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the b. The volumes are the same. The volume of a prism has an area of 10 square units, then its volume pyramid equals one third times the base area times is 10h cubic units. So, the cone must have a base the height. So, if the base areas of two pyramids are area of 30 square units so that its volume is equal and their heights are equal, then their volumes

are equal. or 10h cubic units. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the ANSWER: volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied Sometimes; the statement is true if the base area of by 5 · 5 or 25. the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an 36. CCSS ARGUMENTS Determine whether the area of 10 square units, then its volume is 10h cubic following statement is sometimes, always, or never units. So, the cone must have a base area of 30 true. Justify your reasoning. square units so that its volume is or 10h The volume of a cone with radius r and height h equals the volume of a prism with height h. cubic units. SOLUTION: 37. ERROR ANALYSIS Alexandra and Cornelio are The volume of a cone with a radius r and height h is calculating the volume of the cone below. Is either of . The volume of a prism with a height of them correct? Explain your answer. h is where B is the area of the base of the prism. Set the volumes equal.

The volumes will only be equal when the radius of the cone is equal to or when . Alexandra incorrectly used the slant height. Therefore, the statement is true sometimes if the ANSWER: base area of the cone is 3 times as great as the base Cornelio; Alexandra incorrectly used the slant height. area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume 38. REASONING A cone has a volume of 568 cubic is 10h cubic units. So, the cone must have a base centimeters. What is the volume of a cylinder that area of 30 square units so that its volume is has the same radius and height as the cone? Explain

or 10h cubic units. your reasoning. SOLUTION: ANSWER: 3 1704 cm ; The formula for the volume of a cylinder Sometimes; the statement is true if the base area of is V= Bh, while the formula for the volume of a cone the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an is V = Bh. The volume of a cylinder is three times area of 10 square units, then its volume is 10h cubic as much as the volume of a cone with the same units. So, the cone must have a base area of 30 radius and height. square units so that its volume is or 10h ANSWER: 3 cubic units. 1704 cm ; The volume of a cylinder is three times as much as the volume of a cone with the same radius 37. ERROR ANALYSIS Alexandra and Cornelio are and height. calculating the volume of the cone below. Is either of them correct? Explain your answer. 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and Alexandra incorrectly used the slant height. make the height of the pyramid equal to 3 times the height of the prism. ANSWER: Cornelio; Alexandra incorrectly used the slant height. Sample answer: A square pyramid with a base area of 16 and a 38. REASONING A cone has a volume of 568 cubic height of 12, a prism with a square base area of 16 centimeters. What is the volume of a cylinder that and a height of 4. has the same radius and height as the cone? Explain your reasoning. ANSWER: Sample answer: A square pyramid with a base area SOLUTION: of 16 and a height of 12, a prism with a square base 3 1704 cm ; The formula for the volume of a cylinder area of 16 and a height of 4; if a pyramid and prism is V= Bh, while the formula for the volume of a cone have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as is V = Bh. The volume of a cylinder is three times great as the height of the prism. as much as the volume of a cone with the same radius and height. 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding ANSWER: volumes of prisms and cylinders. 1704 cm3; The volume of a cylinder is three times as SOLUTION: much as the volume of a cone with the same radius and height. To find the volume of each solid, you must know the area of the base and the height. The volume of a 39. OPEN ENDED Give an example of a pyramid and pyramid is one third the volume of a prism that has a prism that have the same base and the same the same height and base area. The volume of a volume. Explain your reasoning. cone is one third the volume of a cylinder that has the same height and base area. SOLUTION: The formula for volume of a prism is V = Bh and the ANSWER: formula for the volume of a pyramid is one-third of To find the volume of each solid, you must know the that. So, if a pyramid and prism have the same base, area of the base and the height. The volume of a then in order to have the same volume, the height of pyramid is one third the volume of a prism that has the pyramid must be 3 times as great as the height of the same height and base area. The volume of a the prism. cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the

height of the prism. A 12π

B 15π Sample answer:

A square pyramid with a base area of 16 and a C height of 12, a prism with a square base area of 16 and a height of 4. D ANSWER: Sample answer: A square pyramid with a base area SOLUTION: of 16 and a height of 12, a prism with a square base Use the Pythagorean Theorem to find the radius r of area of 16 and a height of 4; if a pyramid and prism the cone. have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

ANSWER: So, the radius of the cone is 3 centimeters. To find the volume of each solid, you must know the area of the base and the height. The volume of a The volume of a circular cone is , where pyramid is one third the volume of a prism that has r is the radius of the base and h is the height of the the same height and base area. The volume of a cone. cone is one third the volume of a cylinder that has the

same height and base area. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

Therefore, the correct choice is A.

ANSWER: A 12π A B 15π

C 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, D how tall is the tent’s center pole?

SOLUTION: SOLUTION: Use the Pythagorean Theorem to find the radius r of The volume of a pyramid is , where B is the the cone. area of the base and h is the height of the pyramid.

ANSWER: 5.5 ft

So, the radius of the cone is 3 centimeters. 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows The volume of a circular cone is , where the results of several spins. What is the experimental probability of the spinner landing on orange? r is the radius of the base and h is the height of the cone.

G Therefore, the correct choice is A.

H ANSWER: A J 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is SOLUTION: 6 feet by 8 feet. If the tent holds 88 cubic feet of air, Possible outcomes: {6 red, 4 blue, 5 orange, 10 how tall is the tent’s center pole? green} Number of possible outcomes : 25 SOLUTION: Favorable outcomes: {5 orange} Number of favorable outcomes: 5 The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

So, the correct choice is F.

ANSWER: F

ANSWER: 44. SAT/ACT For all 5.5 ft A –8 43. PROBABILITY A spinner has sections colored B x – 4 red, blue, orange, and green. The table below shows C the results of several spins. What is the experimental probability of the spinner landing on orange? D E

SOLUTION:

G So, the correct choice is E.

H ANSWER: E J Find the volume of each prism. SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} 45. Number of favorable outcomes: 5 SOLUTION:

The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the correct choice is F.

ANSWER: F

3 Therefore, the volume of the prism is 1008 in . 44. SAT/ACT For all ANSWER: A –8 3 B x – 4 1008 in C D E 46. SOLUTION: SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will So, the correct choice is E. bisect the base. Use the Pythagorean Theorem to find the height of the triangle. ANSWER: E Find the volume of each prism.

45.

SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches. So, the height of the triangle is 12 feet. Find the area of the triangle.

3 Therefore, the volume of the prism is 1008 in . So, the area of the base B is 60 ft2. ANSWER: 1008 in3 The height h of the prism is 19 feet.

46. Therefore, the volume of the prism is 1140 ft3.

SOLUTION: ANSWER: The volume of a prism is , where B is the 3 1140 ft area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the height of the triangle is 12 feet. Find the area of the triangle. Therefore, the volume of the prism is about 426,437.6

m3.

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain 2 So, the area of the base B is 60 ft . after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the The height h of the prism is 19 feet. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863.

SOLUTION: Therefore, the volume of the prism is 1140 ft3. To find the entire surface area of the bin, find the surface area of the conical top and bottom and find ANSWER: the surface area of the cylinder and add them. 3 The formula for finding the surface area of a cone is 1140 ft , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

Therefore, the volume of the prism is about 426,437.6

m3.

ANSWER: 3 426,437.6 m

area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the The formula for finding the surface area of a cylinder nearest square foot. is , where is r is the radius and h is the slant Refer to the photo on Page 863. height of the cylinder.

SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find Surface area of the bin = Surface area of the

the surface area of the cylinder and add them. cylinder + Surface area of the conical top + surface The formula for finding the surface area of a cone is area of the conical bottom. , where is r is the radius and l is the slant height

of the cone.

Find the slant height of the conical top.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49.

Find the slant height of the conical bottom. SOLUTION: The height of the conical bottom is 28 – (5 + 12 + 2) Area of the shaded region = Area of the rectangle – or 9 ft. Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: 30.4 cm2

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder. 50.

SOLUTION: Surface area of the bin = Surface area of the Area of the shaded region = Area of the circle – cylinder + Surface area of the conical top + surface Area of the hexagon area of the conical bottom.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the ANSWER: angle, so we use 30° when using trig to find the length of the apothem.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION:

Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: 30.4 cm2

Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the

interior angle is . The apothem bisects the ANSWER: angle, so we use 30° when using trig to find the 2 length of the apothem. 54.4 in

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon. The equilateral triangle can be divided into three

Use a trigonometric ratio to find the value of b. ANSWER: 2 54.4 in

51. Now use the area of the isosceles triangles to find the area of the equilateral triangle. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the equilateral triangle is about 67.3 square feet.

So, the area of the circle is about 40.7 square feet. Subtract to find the area of shaded region.

Next, find the area of the equilateral triangle.

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is

half the length b of the side of the equilateral triangle. 52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed Use a trigonometric ratio to find the value of b. circle.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

Use trigonometric ratios to find the values of b and h.

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft

The area of the equilateral triangle will equal three 52. times the area of any of the isosceles triangles. Find the area of the shaded region. SOLUTION: The area of the shaded region is the area of the A(shaded region) = A(circumscribed circle) - A circumscribed circle minus the area of the equilateral (equilateral triangle) + A(inscribed circle) triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: Divide the equilateral triangle into three isosceles 2 168.2 mm triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION: Find the volume of each pyramid. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3

1. centimeters. The height of the pyramid is 12 SOLUTION: centimeters.

ANSWER: 132 cm3

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION: ANSWER: 3 75 in The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

2. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: The base of this pyramid is a regular pentagon with 3 sides of 4.4 centimeters and an apothem of 3 62.4 m centimeters. The height of the pyramid is 12 4. a square pyramid with a height of 14 meters and a centimeters. base with 8-meter side lengths

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

ANSWER: 132 cm3

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters ANSWER: SOLUTION: 3 298.7 m

The volume of a pyramid is , where B is the Find the volume of each cone. Round to the area of the base and h is the height of the pyramid. nearest tenth. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

5. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the ANSWER: 3 height of the cone, and r is the radius of the base. 62.4 m Since the diameter of this cone is 7 inches, the radius 4. a square pyramid with a height of 14 meters and a is or 3.5 inches. The height of the cone is 4 inches. base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. ANSWER: 3 51.3 in

ANSWER: 6. 298.7 m3 SOLUTION: Find the volume of each cone. Round to the nearest tenth.

Use trigonometry to find the radius r. 5. SOLUTION:

The volume of a circular cone is , or The volume of a circular cone is , or

, where B is the area of the base, h is the , where B is the area of the base, h is the height of the cone, and r is the radius of the base. height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius The height of the cone is 11.5 centimeters. is or 3.5 inches. The height of the cone is 4 inches.

ANSWER: 168.1 cm3 ANSWER: 7. an oblique cone with a height of 10.5 millimeters and 3 51.3 in a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the 6. height of the cone, and r is the radius of the base. SOLUTION: The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

Use trigonometry to find the radius r.

ANSWER: 3 The volume of a circular cone is , or 28.1 mm

, where B is the area of the base, h is the 8. a cone with a slant height of 25 meters and a radius of 15 meters height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. SOLUTION:

Use the Pythagorean Theorem to find the height h of ANSWER: the cone. Then find its volume. 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or So, the height of the cone is 20 meters. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

ANSWER: 4712.4 m3

9. MUSEUMS The sky dome of the National Corvette ANSWER: Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the 3 28.1 mm base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have 8. a cone with a slant height of 25 meters and a radius to accommodate. Round to the nearest tenth. of 15 meters SOLUTION: SOLUTION: The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet. Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

ANSWER: 3 513,333.3 ft So, the height of the cone is 20 meters. CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

ANSWER: 10. 4712.4 m3 SOLUTION: 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical The volume of a pyramid is , where B is the building. If the height is 100 feet and the area of the area of the base and h is the height of the pyramid. base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. ANSWER: For this cone, the area of the base is 15,400 square 605 in3 feet and the height is 100 feet.

11. SOLUTION: ANSWER: 3 513,333.3 ft The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. CCSS SENSE-MAKING Find the volume of

each pyramid. Round to the nearest tenth if necessary.

10. ANSWER: 3 SOLUTION: 105.8 mm

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 605 in3

ANSWER: 11. 482.1 m3 SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid. ANSWER: 3 The base is a hexagon, so we need to make a right tri 105.8 mm determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

ANSWER: 482.1 m3

13. ANSWER: 3 SOLUTION: 233.8 cm

The volume of a pyramid is , where B is the 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet base and h is the height of the pyramid. SOLUTION: The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he The volume of a pyramid is , where B is the 120°. The radius bisects the angle, so the right triangl area of the base and h is the height of the pyramid. 90° triangle.

ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters The apothem is . SOLUTION: Find the height of the right triangle.

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters ANSWER: SOLUTION: 3 Find the height of the right triangle. 35.6 cm 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The length of the other leg of the right triangle is 6 cm.

ANSWER:

3 35.6 cm So, the area of the base B is 24 cm2.

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has Replace V with 144 and B with 24 in the formula for

a volume of 144 cubic centimeters. Find the height. the volume of a pyramid and solve for the height h.

SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

Therefore, the height of the triangular pyramid is 18 cm.

ANSWER: 18 cm

Find the volume of each cone. Round to the nearest tenth.

17. The length of the other leg of the right triangle is 6 cm. SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 So, the area of the base B is 24 cm2. inches.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 3 235.6 in Therefore, the height of the triangular pyramid is 18 cm.

ANSWER: 18 cm 18. Find the volume of each cone. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and 17. the height is 7.3 centimeters. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 Therefore, the volume of the cone is about 134.8 3 inches. cm .

ANSWER:

134.8 cm3

3 Therefore, the volume of the cone is about 235.6 in . 19. ANSWER: SOLUTION: 235.6 in3

18. SOLUTION: Use a trigonometric ratio to find the height h of the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Therefore, the volume of the cone is about 134.8 3 cm . ANSWER: 134.8 cm3 Therefore, the volume of the cone is about 1473.1 3 cm .

ANSWER: 1473.1 cm3

19. SOLUTION:

20. SOLUTION:

Use a trigonometric ratio to find the height h of the

cone. Use trigonometric ratios to find the height h and the radius r of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 1473.1 3 cm .

ANSWER: 3 1473.1 cm 3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

20. 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION: SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches. Use trigonometric ratios to find the height h and the radius r of the cone.

Therefore, the volume of the cone is about 1072.3 3 The volume of a circular cone is , where in . r is the radius of the base and h is the height of the ANSWER: cone. 1072.3 in3

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

3 Therefore, the volume of the cone is about 1072.3 Therefore, the volume of the cone is about 5.8 cm . 3 in . ANSWER: ANSWER: 5.8 cm3 1072.3 in3 23. SNACKS Approximately how many cubic 22. a right cone with a slant height of 5.6 centimeters centimeters of roasted peanuts will completely fill a and a radius of 1 centimeter paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest SOLUTION: tenth. The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean SOLUTION: Theorem to find the height h of the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. ANSWER: 234.6 cm3

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its 3 Therefore, the volume of the cone is about 5.8 cm . square base is 600 feet wide. Find the volume of this pyramid. ANSWER: SOLUTION: 5.8 cm3 The volume of a pyramid is , where B is the 23. SNACKS Approximately how many cubic

centimeters of roasted peanuts will completely fill a area of the base and h is the height of the pyramid. paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the ANSWER: cone. Since the diameter of the cone is 8 3 42,000,000 ft centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters. 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: SOLUTION: 234.6 cm3 The volume of a pyramid is , where B is the 24. CCSS MODELING The Pyramid Arena in area of the base and h is the height of the pyramid. Memphis, Tennessee, is the third largest pyramid in The base of the pyramid is a regular octagon with the world. It is approximately 350 feet tall, and its sides of 2 feet. A central angle of the octagon is square base is 600 feet wide. Find the volume of this pyramid. or 45°, so the angle formed in the triangle below is 22.5°. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Use a trigonometric ratio to find the apothem a.

ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular The height of this pyramid is 5 feet. octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Therefore, the volume of the greenhouse is about SOLUTION: 32.2 ft3. The volume of a pyramid is , where B is the ANSWER: area of the base and h is the height of the pyramid. 3 The base of the pyramid is a regular octagon with 32.2 ft sides of 2 feet. A central angle of the octagon is Find the volume of each solid. Round to the or 45°, so the angle formed in the triangle nearest tenth. below is 22.5°.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet.

ANSWER: 471.2 in3

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 27. 3 32.2 ft SOLUTION: Find the volume of each solid. Round to the nearest tenth.

26.

SOLUTION: ANSWER: Volume of the solid given = Volume of the small 3 cone + Volume of the large cone 3190.6 m

28. SOLUTION:

ANSWER: 471.2 in3

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her 27. backyard. To buy a heating unit for the space, she SOLUTION: needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

ANSWER: 3 3190.6 m SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

28. SOLUTION:

The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = ANSWER: 3 6666.67 ft . Two BTU's are needed for every cubic 3 7698.5 cm foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) ANSWER: required to heat the building. For new construction 13,333 BTUs with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to 30. SCIENCE Marta is studying crystals that grow on purchase? rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches.

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION: SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base The volume of a pyramid is , where B is the is area of the base and h is the height of the pyramid.

The volume of the ceiling is

It tells Marta how much clay is needed to make the model.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model. The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic 31. CHANGING DIMENSIONS A cone has a radius foot, so the size of the heating unit Sam should buy is of 4 centimeters and a height of 9 centimeters. 6666.67 × 2 = 13,333 BTUs. Describe how each change affects the volume of the cone. a. The height is doubled. ANSWER: b. The radius is doubled. 13,333 BTUs c. Both the radius and the height are doubled. 30. SCIENCE Marta is studying crystals that grow on SOLUTION: rock formations.For a project, she is making a clay Find the volume of the original cone. Then alter the model of a crystal with a shape that is a composite of values. two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches.

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the a. Double h. area of the base and h is the height of the pyramid.

The volume is doubled. It tells Marta how much clay is needed to make the model. b. Double r.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. 2 Describe how each change affects the volume of the The volume is multiplied by 2 or 4. cone. a. The height is doubled. c. Double r and h. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

volume is multiplied by 23 or 8.

ANSWER: a. The volume is doubled. 2 12-5 Volumes of Pyramids and Cones b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8. a. Double h. Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume is doubled. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. b. Double r. Let s be the side length of the base.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

The side length of the base is 15 cm.

ANSWER:

15 cm volume is multiplied by 23 or 8.

ANSWER: 33. The volume of a cone is 196π cubic inches and the a. The volume is doubled. height is 12 inches. What is the diameter? 2 b. The volume is multiplied by 2 or 4. SOLUTION: c. The volume is multiplied by 23 or 8. The volume of a circular cone is , or Find each measure. Round to the nearest tenth if necessary. , where B is the area of the base, h is the 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find height of the cone, and r is the radius of the base. the side length of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

eSolutions Manual - Powered by Cognero Page 13

The diameter is 2(7) or 14 inches.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the The side length of the base is 15 cm. volume of the cone?

ANSWER: SOLUTION: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the The lateral area of a cone is , where r is the radius and is the slant height of the cone. height of the cone, and r is the radius of the base. Replace L with 71.6 and with 6, then solve for the Since the diameter is 8 centimeters, the radius is 4 radius r. centimeters.

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

The diameter is 2(7) or 14 inches.

ANSWER: 14 in.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the So, the height of the cone is about 4.64 millimeters. volume of the cone? SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The lateral area of a cone is , where r is

the radius and is the slant height of the cone. Therefore, the volume of the cone is about 70.2 Replace L with 71.6 and with 6, then solve for the 3 radius r. mm .

ANSWER:

70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with So, the radius is about 3.8 millimeters. different bases that have a height of 10 centimeters and a base area of 24 square centimeters. Use the Pythagorean Theorem to find the height of b. VERBAL What is true about the volumes of the the cone. two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

So, the height of the cone is about 4.64 millimeters. Sample answer:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3 b. The volumes are the same. The volume of a 35. MULTIPLE REPRESENTATIONS In this pyramid equals one third times the base area times problem, you will investigate rectangular pyramids. the height. So, if the base areas of two pyramids are a. GEOMETRIC Draw two pyramids with equal and their heights are equal, then their volumes different bases that have a height of 10 centimeters are equal. and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

ANSWER: c. If the base area is multiplied by 5, the volume is a. Sample answer: multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h ANSWER: equals the volume of a prism with height h. a. Sample answer: SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

b. The volumes are the same. The volume of a The volumes will only be equal when the radius of pyramid equals one third times the base area times the cone is equal to or when . the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes Therefore, the statement is true sometimes if the are equal. base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the c. If the base area is multiplied by 5, the volume is prism has an area of 10 square units, then its volume multiplied by 5. If the height is multiplied by 5, the is 10h cubic units. So, the cone must have a base volume is multiplied by 5. If both the base area and area of 30 square units so that its volume is the height are multiplied by 5, the volume is multiplied

by 5 · 5 or 25. or 10h cubic units. 36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never ANSWER: true. Justify your reasoning. Sometimes; the statement is true if the base area of The volume of a cone with radius r and height h the cone is 3 times as great as the base area of the equals the volume of a prism with height h. prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic SOLUTION: units. So, the cone must have a base area of 30 The volume of a cone with a radius r and height h is square units so that its volume is or 10h . The volume of a prism with a height of

h is where B is the area of the base of the cubic units. prism. Set the volumes equal. 37. ERROR ANALYSIS Alexandra and Cornelio are

calculating the volume of the cone below. Is either of them correct? Explain your answer.

SOLUTION: The volumes will only be equal when the radius of The slant height is used for surface area, but the the cone is equal to or when . height is used for volume. For this cone, the slant Therefore, the statement is true sometimes if the height of 13 is provided, and we need to calculate the base area of the cone is 3 times as great as the base height before we can calculate the volume. area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base Alexandra incorrectly used the slant height. area of 30 square units so that its volume is ANSWER:

or 10h cubic units. Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic ANSWER: centimeters. What is the volume of a cylinder that Sometimes; the statement is true if the base area of has the same radius and height as the cone? Explain the cone is 3 times as great as the base area of the your reasoning. prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic SOLUTION: units. So, the cone must have a base area of 30 3 1704 cm ; The formula for the volume of a cylinder square units so that its volume is or 10h is V= Bh, while the formula for the volume of a cone cubic units. is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same 37. ERROR ANALYSIS Alexandra and Cornelio are radius and height. calculating the volume of the cone below. Is either of ANSWER: them correct? Explain your answer. 1704 cm3; The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. SOLUTION: The formula for volume of a prism is V = Bh and the

formula for the volume of a pyramid is one-third of SOLUTION: that. So, if a pyramid and prism have the same base, The slant height is used for surface area, but the then in order to have the same volume, the height of height is used for volume. For this cone, the slant the pyramid must be 3 times as great as the height of height of 13 is provided, and we need to calculate the the prism. height before we can calculate the volume.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height. Set the base areas of the prism and pyramid, and 38. REASONING A cone has a volume of 568 cubic make the height of the pyramid equal to 3 times the centimeters. What is the volume of a cylinder that height of the prism. has the same radius and height as the cone? Explain your reasoning. Sample answer: A square pyramid with a base area of 16 and a SOLUTION: height of 12, a prism with a square base area of 16 3 1704 cm ; The formula for the volume of a cylinder and a height of 4. is V= Bh, while the formula for the volume of a cone ANSWER: is V = Bh. The volume of a cylinder is three times Sample answer: A square pyramid with a base area as much as the volume of a cone with the same of 16 and a height of 12, a prism with a square base

radius and height. area of 16 and a height of 4; if a pyramid and prism ANSWER: have the same base, then in order to have the same 3 volume, the height of the pyramid must be 3 times as 1704 cm ; The volume of a cylinder is three times as great as the height of the prism. much as the volume of a cone with the same radius and height. 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding 39. OPEN ENDED Give an example of a pyramid and volumes of prisms and cylinders. a prism that have the same base and the same volume. Explain your reasoning. SOLUTION: To find the volume of each solid, you must know the SOLUTION: area of the base and the height. The volume of a The formula for volume of a prism is V = Bh and the pyramid is one third the volume of a prism that has formula for the volume of a pyramid is one-third of the same height and base area. The volume of a that. So, if a pyramid and prism have the same base, cone is one third the volume of a cylinder that has the then in order to have the same volume, the height of same height and base area. the pyramid must be 3 times as great as the height of the prism. ANSWER: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

Set the base areas of the prism and pyramid, and 41. A conical sand toy has the dimensions as shown make the height of the pyramid equal to 3 times the below. How many cubic centimeters of sand will it height of the prism. hold when it is filled to the top?

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4. A 12π ANSWER: B 15π

Sample answer: A square pyramid with a base area C of 16 and a height of 12, a prism with a square base area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same D volume, the height of the pyramid must be 3 times as great as the height of the prism. SOLUTION: Use the Pythagorean Theorem to find the radius r of 40. WRITING IN MATH Compare and contrast the cone. finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

ANSWER: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the So, the radius of the cone is 3 centimeters. same height and base area. The volume of a circular cone is , where 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it r is the radius of the base and h is the height of the

hold when it is filled to the top? cone.

A 12π B 15π

C Therefore, the correct choice is A. ANSWER: D A

SOLUTION: 42. SHORT RESPONSE Brooke is buying a tent that Use the Pythagorean Theorem to find the radius r of is in the shape of a rectangular pyramid. The base is the cone. 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the radius of the cone is 3 centimeters.

The volume of a circular cone is , where ANSWER: r is the radius of the base and h is the height of the 5.5 ft cone. 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows

the results of several spins. What is the experimental probability of the spinner landing on orange?

Therefore, the correct choice is A.

ANSWER: F A G 42. SHORT RESPONSE Brooke is buying a tent that

is in the shape of a rectangular pyramid. The base is H 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? J SOLUTION: SOLUTION: The volume of a pyramid is , where B is the Possible outcomes: {6 red, 4 blue, 5 orange, 10 area of the base and h is the height of the pyramid. green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

ANSWER: So, the correct choice is F. 5.5 ft ANSWER: 43. PROBABILITY A spinner has sections colored F red, blue, orange, and green. The table below shows the results of several spins. What is the experimental 44. SAT/ACT For all probability of the spinner landing on orange? A –8 B x – 4 C

F E

G SOLUTION:

J So, the correct choice is E. SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 ANSWER: green} E Number of possible outcomes : 25 Favorable outcomes: {5 orange} Find the volume of each prism. Number of favorable outcomes: 5

45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The So, the correct choice is F. base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the ANSWER: prism is 6 inches. F

44. SAT/ACT For all

A –8 B x – 4 C 3 Therefore, the volume of the prism is 1008 in . D ANSWER: E 1008 in3

SOLUTION:

46. So, the correct choice is E. SOLUTION: ANSWER: The volume of a prism is , where B is the area of the base and h is the height of the prism. The E base of this prism is an isosceles triangle with a base Find the volume of each prism. of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the height of the triangle is 12 feet. Find the area of the triangle.

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 1008 in3

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base Therefore, the volume of the prism is 1140 ft3. of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to ANSWER: find the height of the triangle. 3 1140 ft

47.

SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the

prism is 102.3 meters. So, the height of the triangle is 12 feet. Find the area

of the triangle.

Therefore, the volume of the prism is about 426,437.6 m3.

So, the area of the base B is 60 ft2. ANSWER: 3 426,437.6 m The height h of the prism is 19 feet. 48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write Therefore, the volume of the prism is 1140 ft3. the exact answer and the answer rounded to the nearest square foot. ANSWER: Refer to the photo on Page 863. 3 1140 ft SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height 47. of the cone.

SOLUTION: Find the slant height of the conical top. The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6

3 m . Find the slant height of the conical bottom. ANSWER: The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the The formula for finding the surface area of a cylinder surface area of the conical top and bottom and find is , where is r is the radius and h is the slant the surface area of the cylinder and add them. height of the cylinder. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. Surface area of the bin = Surface area of the Find the slant height of the conical top. cylinder + Surface area of the conical top + surface area of the conical bottom.

ANSWER:

Find the area of each shaded region. Polygons Find the slant height of the conical bottom. in 50 - 52 are regular. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

49.

SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant ANSWER: height of the cylinder. 2 30.4 cm

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom. 50.

SOLUTION:

Area of the shaded region = Area of the circle – Area of the hexagon

ANSWER:

A regular hexagon has 6 sides, so the measure of the Find the area of each shaded region. Polygons in 50 - 52 are regular. interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem. 49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

ANSWER:

2 30.4 cm

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon Now find the area of the hexagon.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

ANSWER: 2 54.4 in

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

ANSWER: 2 54.4 in

Use a trigonometric ratio to find the value of b.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. Now use the area of the isosceles triangles to find Find the area of the circle with a radius of 3.6 feet. the area of the equilateral triangle.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle. So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

The equilateral triangle can be divided into three Therefore, the area of the shaded region is about 26.6 ft2. isosceles triangles with central angles of or 120°, so the angle in the triangle created by the ANSWER: height of the isosceles triangle is 60° and the base is 2 half the length b of the side of the equilateral triangle. 26.6 ft

52. SOLUTION: Use a trigonometric ratio to find the value of b. The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The So, the area of the equilateral triangle is about 67.3 angle in the triangle formed by the height.h of the square feet. isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown Subtract to find the area of shaded region. below.

Therefore, the area of the shaded region is about 26.6 ft2. Use trigonometric ratios to find the values of b and ANSWER: h. 2 26.6 ft

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral The area of the equilateral triangle will equal three triangle plus the area of the inscribed circle. Let b times the area of any of the isosceles triangles. Find represent the length of each side of the equilateral the area of the shaded region. triangle and h represent the radius of the inscribed circle. A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Divide the equilateral triangle into three isosceles Therefore, the area of the shaded region is about triangles with central angles of or 120°. The 168.2 mm2. angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown ANSWER: 2 below. 168.2 mm

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

Find the volume of each pyramid.

ANSWER: 3 75 in ANSWER: 3 62.4 m

area of the base and h is the height of the pyramid. meters. The height of the pyramid is 14 meters. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth.

ANSWER: 132 cm3 5. 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

ANSWER: ANSWER: 3 3 62.4 m 51.3 in 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

Use trigonometry to find the radius r.

ANSWER: 298.7 m3 The volume of a circular cone is , or

Find the volume of each cone. Round to the , where B is the area of the base, h is the nearest tenth. height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

5. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. ANSWER: 3 51.3 in

6. ANSWER: 3 28.1 mm SOLUTION: 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

Use trigonometry to find the radius r.

The volume of a circular cone is , or Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. , where B is the area of the base, h is the

height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

So, the height of the cone is 20 meters.

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: ANSWER: 3 The volume of a circular cone is , or 4712.4 m 9. MUSEUMS The sky dome of the National Corvette , where B is the area of the base, h is the Museum in Bowling Green, Kentucky, is a conical height of the cone, and r is the radius of the base. building. If the height is 100 feet and the area of the The radius of this cone is 1.6 millimeters and the base is about 15,400 square feet, find the volume of height is 10.5 millimeters. air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of Use the Pythagorean Theorem to find the height h of each pyramid. Round to the nearest tenth if the cone. Then find its volume. necessary.

10. SOLUTION: So, the height of the cone is 20 meters. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: 4712.4 m3 ANSWER: MUSEUMS 9. The sky dome of the National Corvette 3 Museum in Bowling Green, Kentucky, is a conical 605 in building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B 11. is the area of the base and h is the height of the SOLUTION: cone. The volume of a pyramid is , where B is the For this cone, the area of the base is 15,400 square feet and the height is 100 feet. area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 3 3 513,333.3 ft 105.8 mm

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

12. SOLUTION:

The volume of a pyramid is , where B is the 10. area of the base and h is the height of the pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 482.1 m3

ANSWER: 605 in3

13. SOLUTION:

ANSWER: 3 105.8 mm

The apothem is . 12.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 233.8 cm

ANSWER: 14. a pentagonal pyramid with a base area of 590 square 3 482.1 m feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid. ANSWER: The base is a hexagon, so we need to make a right tri 3 determine the apothem. The interior angles of the he 1376.7 ft 120°. The radius bisects the angle, so the right triangl 15. a triangular pyramid with a height of 4.8 centimeters

90° triangle. and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has

ANSWER: a volume of 144 cubic centimeters. Find the height. 1376.7 ft3 SOLUTION: The base of the pyramid is a right triangle with a leg 15. a triangular pyramid with a height of 4.8 centimeters of 8 centimeters and a hypotenuse of 10 centimeters. and a right triangle base with a leg 5 centimeters and Use the Pythagorean Theorem to find the other leg a hypotenuse 10.2 centimeters of the right triangle and then find the area of the triangle. SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a Therefore, the height of the triangular pyramid is 18 leg 8 centimeters and hypotenuse 10 centimeters has cm. a volume of 144 cubic centimeters. Find the height. SOLUTION: ANSWER: The base of the pyramid is a right triangle with a leg 18 cm of 8 centimeters and a hypotenuse of 10 centimeters. Find the volume of each cone. Round to the Use the Pythagorean Theorem to find the other leg a nearest tenth. of the right triangle and then find the area of the triangle.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: So, the area of the base B is 24 cm2. 235.6 in3

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

18. SOLUTION:

Therefore, the volume of the cone is about 134.8 3 cm .

17. ANSWER: SOLUTION: 134.8 cm3 The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the 19. radius is or 5 inches. The height of the cone is 9 SOLUTION: inches.

Use a trigonometric ratio to find the height h of the 3 Therefore, the volume of the cone is about 235.6 in . cone.

ANSWER: 235.6 in3

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. 18. SOLUTION:

ANSWER: 1473.1 cm3

Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 20. 134.8 cm3 SOLUTION:

19. Use trigonometric ratios to find the height h and the SOLUTION: radius r of the cone.

Use a trigonometric ratio to find the height h of the The volume of a circular cone is , where cone. r is the radius of the base and h is the height of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an

Therefore, the volume of the cone is about 1473.1 altitude of 16 inches 3 cm . SOLUTION:

ANSWER: The volume of a circular cone is , where 3 1473.1 cm r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

20. SOLUTION:

Therefore, the volume of the cone is about 1072.3 3 in . ANSWER: Use trigonometric ratios to find the height h and the 3 radius r of the cone. 1072.3 in 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter

SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER:

3 2.8 ft 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

the radius is or 8 inches. 5.8 cm3

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION: Therefore, the volume of the cone is about 1072.3 3 The volume of a circular cone is , where in . r is the radius of the base and h is the height of the ANSWER: cone. Since the diameter of the cone is 8 1072.3 in3 centimeters, the radius is or 4 centimeters. The 22. a right cone with a slant height of 5.6 centimeters height of the cone is 14 centimeters. and a radius of 1 centimeter

SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 3 ANSWER: 5.8 cm 3 42,000,000 ft 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a 25. GARDENING The greenhouse is a regular paper cone that is 14 centimeters high and has a base octagonal pyramid with a height of 5 feet. The base diameter of 8 centimeters? Round to the nearest has side lengths of 2 feet. What is the volume of the tenth. greenhouse? SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The

height of the cone is 14 centimeters. SOLUTION:

3 below is 22.5°. Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The height of this pyramid is 5 feet.

ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular Therefore, the volume of the greenhouse is about octagonal pyramid with a height of 5 feet. The base 32.2 ft3. has side lengths of 2 feet. What is the volume of the greenhouse? ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

Use a trigonometric ratio to find the apothem a. ANSWER: 471.2 in3

The height of this pyramid is 5 feet.

27. SOLUTION:

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: ANSWER: 3 3 32.2 ft 3190.6 m Find the volume of each solid. Round to the nearest tenth.

28. 26. SOLUTION: SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) ANSWER: required to heat the building. For new construction 3 with good insulation, there should be 2 BTUs per 471.2 in cubic foot. What size unit does Sam need to purchase?

27.

SOLUTION: SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

ANSWER: The volume of the ceiling is 3 3190.6 m

28. The total volume is therefore 5000 + 1666.67 = 3 SOLUTION: 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay

model of a crystal with a shape that is a composite of ANSWER: two congruent rectangular pyramids. The base of 3 each pyramid will be 1 by 1.5 inches, and the total 7698.5 cm height will be 4 inches.

HEATING 29. Sam is building an art studio in her Determine the volume of the model. Explain why backyard. To buy a heating unit for the space, she knowing the volume is helpful in this situation. needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction SOLUTION: with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to The volume of a pyramid is , where B is the purchase? area of the base and h is the height of the pyramid.

ANSWER:

13,333 BTUs 30. SCIENCE Marta is studying crystals that grow on a. Double h. rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches.

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. The volume is doubled.

SOLUTION: b. Double r. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

2 The volume is multiplied by 2 or 4.

It tells Marta how much clay is needed to make the c. Double r and h. model.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. volume is multiplied by 23 or 8. Describe how each change affects the volume of the cone. ANSWER: a. The height is doubled. a. The volume is doubled. 2 b. The radius is doubled. b. The volume is multiplied by 2 or 4. c. Both the radius and the height are doubled. 3 c. The volume is multiplied by 2 or 8. SOLUTION: Find each measure. Round to the nearest tenth Find the volume of the original cone. Then alter the if necessary. values. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base. a. Double h.

The volume is doubled.

b. Double r.

The side length of the base is 15 cm.

ANSWER: 15 cm

2 The volume is multiplied by 2 or 4. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? c. Double r and h. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base.

3 Since the diameter is 8 centimeters, the radius is 4 volume is multiplied by 2 or 8. centimeters.

ANSWER:

a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION: The diameter is 2(7) or 14 inches. The volume of a pyramid is , where B is the ANSWER: area of the base and h is the height of the pyramid. Let s be the side length of the base. 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The side length of the base is 15 cm. The lateral area of a cone is , where r is the radius and is the slant height of the cone. ANSWER: 12-5 Volumes of Pyramids and Cones Replace L with 71.6 and with 6, then solve for the 15 cm radius r.

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or So, the radius is about 3.8 millimeters. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Use the Pythagorean Theorem to find the height of Since the diameter is 8 centimeters, the radius is 4 the cone. centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the

The diameter is 2(7) or 14 inches. cone.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone?

SOLUTION: Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters The lateral area of a cone is , where r is and a base area of 24 square centimeters. the radius and is the slant height of the cone. b. VERBAL What is true about the volumes of the Replace L with 71.6 and with 6, then solve for the two pyramids that you drew? Explain.

radius r. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that eSolutions Manual - Powered by Cognero multiply to make 24. Page 14

So, the radius is about 3.8 millimeters. Sample answer:

Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER:

70.2 mm3

b. VERBAL What is true about the volumes of the c. two pyramids that you drew? Explain. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the c. ANALYTICAL Explain how multiplying the base volume is multiplied by 5. If both the base area and area and/or the height of the pyramid by 5 affects the the height are multiplied by 5, the volume is multiplied volume of the pyramid. by 5 · 5 or 25. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

ANSWER: a. Sample answer:

or 10h cubic units.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic The volumes will only be equal when the radius of centimeters. What is the volume of a cylinder that the cone is equal to or when . has the same radius and height as the cone? Explain Therefore, the statement is true sometimes if the your reasoning. base area of the cone is 3 times as great as the base SOLUTION: area of the prism. For example, if the base of the 3 prism has an area of 10 square units, then its volume 1704 cm ; The formula for the volume of a cylinder is 10h cubic units. So, the cone must have a base is V= Bh, while the formula for the volume of a cone area of 30 square units so that its volume is is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same or 10h cubic units. radius and height.

A 12π B 15π

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders.

SOLUTION: To find the volume of each solid, you must know the So, the radius of the cone is 3 centimeters. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has The volume of a circular cone is , where the same height and base area. The volume of a r is the radius of the base and h is the height of the cone is one third the volume of a cylinder that has the cone.

41. A conical sand toy has the dimensions as shown ANSWER: below. How many cubic centimeters of sand will it A hold when it is filled to the top? 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? A 12π SOLUTION: B 15π The volume of a pyramid is , where B is the C area of the base and h is the height of the pyramid.

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where

r is the radius of the base and h is the height of the F cone. G

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. So, the correct choice is F.

ANSWER: F

44. SAT/ACT For all

A –8 B x – 4 C D ANSWER: 5.5 ft E

So, the correct choice is E.

ANSWER: E F Find the volume of each prism.

3 Therefore, the volume of the prism is 1008 in .

So, the correct choice is F. ANSWER: 3 ANSWER: 1008 in F

44. SAT/ACT For all

A –8

B x – 4 46. C SOLUTION: D The volume of a prism is , where B is the area of the base and h is the height of the prism. The E base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will SOLUTION: bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the correct choice is E.

ANSWER: E

Find the volume of each prism.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 3 1008 in3 Therefore, the volume of the prism is 1140 ft . ANSWER: 3 1140 ft

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The 47. base of this prism is an isosceles triangle with a base SOLUTION: of 10 feet and two legs of 13 feet. The height h will The volume of a prism is , where B is the bisect the base. Use the Pythagorean Theorem to area of the base and h is the height of the prism. The find the height of the triangle. base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

nearest square foot. Refer to the photo on Page 863. So, the area of the base B is 60 ft2. SOLUTION: The height h of the prism is 19 feet. To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

3 Therefore, the volume of the prism is 1140 ft . Find the slant height of the conical top.

ANSWER: 3 1140 ft

47. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The Find the slant height of the conical bottom. base of this prism is a rectangle with a length of 79.4 The height of the conical bottom is 28 – (5 + 12 + 2) meters and a width of 52.5 meters. The height of the or 9 ft. prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

nearest square foot. area of the conical bottom.

Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height ANSWER:

of the cone.

Find the slant height of the conical top. Find the area of each shaded region. Polygons

in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

ANSWER: 30.4 cm2

50. SOLUTION:

Area of the shaded region = Area of the circle – The formula for finding the surface area of a cylinder Area of the hexagon is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface A regular hexagon has 6 sides, so the measure of the area of the conical bottom. interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION:

Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: Now find the area of the hexagon. 2 30.4 cm

50.

SOLUTION:

Area of the shaded region = Area of the circle – Area of the hexagon

ANSWER: 2 54.4 in

A regular hexagon has 6 sides, so the measure of the

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b.

ANSWER: Now use the area of the isosceles triangles to find 2 the area of the equilateral triangle. 54.4 in

51. SOLUTION: So, the area of the equilateral triangle is about 67.3 The shaded area is the area of the equilateral triangle square feet. less the area of the inscribed circle. Subtract to find the area of shaded region. Find the area of the circle with a radius of 3.6 feet.

Therefore, the area of the shaded region is about 2 So, the area of the circle is about 40.7 square feet. 26.6 ft .

Next, find the area of the equilateral triangle. ANSWER: 2 26.6 ft

52. SOLUTION: The equilateral triangle can be divided into three The area of the shaded region is the area of the isosceles triangles with central angles of or circumscribed circle minus the area of the equilateral 120°, so the angle in the triangle created by the triangle plus the area of the inscribed circle. Let b height of the isosceles triangle is 60° and the base is represent the length of each side of the equilateral half the length b of the side of the equilateral triangle. triangle and h represent the radius of the inscribed

circle.

below. Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and So, the area of the equilateral triangle is about 67.3 h. square feet.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 2 26.6 ft . The area of the equilateral triangle will equal three ANSWER: times the area of any of the isosceles triangles. Find

2 the area of the shaded region. 26.6 ft A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

Find the volume of each pyramid.

1. SOLUTION:

The volume of a pyramid is , where B is the ANSWER: 3 area of the base and h is the height of the pyramid. 132 cm The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the 3. a rectangular pyramid with a height of 5.2 meters pyramid is 10 inches. and a base 8 meters by 4.5 meters SOLUTION:

ANSWER: 3 75 in

ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a 2. base with 8-meter side lengths SOLUTION: SOLUTION:

The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with The base of this pyramid is a square with sides of 8 sides of 4.4 centimeters and an apothem of 3 meters. The height of the pyramid is 14 meters. centimeters. The height of the pyramid is 12 centimeters.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the ANSWER: nearest tenth. 132 cm3

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters 5. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a circular cone is , or area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of , where B is the area of the base, h is the the pyramid is 5.2 meters. height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

ANSWER: 3 62.4 m ANSWER: 4. a square pyramid with a height of 14 meters and a 3 base with 8-meter side lengths 51.3 in SOLUTION:

Use trigonometry to find the radius r. ANSWER: 298.7 m3

Find the volume of each cone. Round to the The volume of a circular cone is , or nearest tenth. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. 5. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the ANSWER:

height of the cone, and r is the radius of the base. 3 Since the diameter of this cone is 7 inches, the radius 168.1 cm is or 3.5 inches. The height of the cone is 4 inches. 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters

SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. ANSWER: The radius of this cone is 1.6 millimeters and the 3 51.3 in height is 10.5 millimeters.

SOLUTION: ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters

Use trigonometry to find the radius r. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

So, the height of the cone is 20 meters. ANSWER:

3 168.1 cm 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or ANSWER: 3 , where B is the area of the base, h is the 4712.4 m height of the cone, and r is the radius of the base. 9. MUSEUMS The sky dome of the National Corvette The radius of this cone is 1.6 millimeters and the Museum in Bowling Green, Kentucky, is a conical height is 10.5 millimeters. building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the ANSWER: cone. 3 28.1 mm For this cone, the area of the base is 15,400 square feet and the height is 100 feet. 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

ANSWER: 3 Use the Pythagorean Theorem to find the height h of 513,333.3 ft the cone. Then find its volume. CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

So, the height of the cone is 20 meters. 10.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 4712.4 m3

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the ANSWER: base is about 15,400 square feet, find the volume of 605 in3 air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B

is the area of the base and h is the height of the 11. cone. For this cone, the area of the base is 15,400 square SOLUTION: feet and the height is 100 feet. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 513,333.3 ft ANSWER: 3 CCSS SENSE-MAKING Find the volume of 105.8 mm each pyramid. Round to the nearest tenth if necessary.

12. 10. SOLUTION:

SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

area of the base and h is the height of the pyramid.

ANSWER: 3 ANSWER: 482.1 m 605 in3

13.

11. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

area of the base and h is the height of the pyramid. The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 3 105.8 mm

12. SOLUTION: The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 3 233.8 cm 482.1 m3 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the 13. area of the base and h is the height of the pyramid.

SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he ANSWER: 120°. The radius bisects the angle, so the right triangl 90° triangle. 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The apothem is .

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

ANSWER: 16. A triangular pyramid with a right triangle base with a 3 leg 8 centimeters and hypotenuse 10 centimeters has 1376.7 ft a volume of 144 cubic centimeters. Find the height. 15. a triangular pyramid with a height of 4.8 centimeters SOLUTION: and a right triangle base with a leg 5 centimeters and The base of the pyramid is a right triangle with a leg hypotenuse 10.2 centimeters of 8 centimeters and a hypotenuse of 10 centimeters. SOLUTION: Use the Pythagorean Theorem to find the other leg a Find the height of the right triangle. of the right triangle and then find the area of the triangle.

The length of the other leg of the right triangle is 6 cm.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. Therefore, the height of the triangular pyramid is 18 SOLUTION: cm. The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. ANSWER: Use the Pythagorean Theorem to find the other leg a 18 cm of the right triangle and then find the area of the triangle. Find the volume of each cone. Round to the nearest tenth.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 The length of the other leg of the right triangle is 6 inches. cm.

3 So, the area of the base B is 24 cm2. Therefore, the volume of the cone is about 235.6 in . ANSWER: Replace V with 144 and B with 24 in the formula for 3 the volume of a pyramid and solve for the height h. 235.6 in

18. SOLUTION:

The volume of a circular cone is , where Therefore, the height of the triangular pyramid is 18 cm. r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and ANSWER: the height is 7.3 centimeters. 18 cm

Find the volume of each cone. Round to the nearest tenth.

17. Therefore, the volume of the cone is about 134.8 3 cm . SOLUTION: ANSWER: The volume of a circular cone is , 134.8 cm3 where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches. 19.

SOLUTION:

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: Use a trigonometric ratio to find the height h of the 3 cone. 235.6 in

The volume of a circular cone is , where 18. r is the radius of the base and h is the height of the SOLUTION: cone. The radius of this cone is 8 centimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the volume of the cone is about 1473.1 3 cm .

ANSWER: 1473.1 cm3 Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 134.8 cm3 20. SOLUTION:

19. SOLUTION:

Use trigonometric ratios to find the height h and the radius r of the cone.

Use a trigonometric ratio to find the height h of the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3 Therefore, the volume of the cone is about 1473.1 3 cm . 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches ANSWER: SOLUTION: 1473.1 cm3 The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches. 20. SOLUTION:

Therefore, the volume of the cone is about 1072.3 3 Use trigonometric ratios to find the height h and the in . radius r of the cone. ANSWER:

1072.3 in3

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant The volume of a circular cone is , where height of 5.6 centimeters. Use the Pythagorean

r is the radius of the base and h is the height of the Theorem to find the height h of the cone. cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, 3 Therefore, the volume of the cone is about 5.8 cm . the radius is or 8 inches. ANSWER: 5.8 cm3

ANSWER: The volume of a circular cone is , where 3 1072.3 in r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters. SOLUTION:

The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 3 234.6 cm

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3

23. SNACKS Approximately how many cubic ANSWER: centimeters of roasted peanuts will completely fill a 3 42,000,000 ft paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest 25. GARDENING The greenhouse is a regular tenth. octagonal pyramid with a height of 5 feet. The base SOLUTION: has side lengths of 2 feet. What is the volume of the greenhouse? The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with

3 sides of 2 feet. A central angle of the octagon is Therefore, the paper cone will hold about 234.6 cm or 45°, so the angle formed in the triangle of roasted peanuts. below is 22.5°. ANSWER: 234.6 cm3

The height of this pyramid is 5 feet.

ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the Therefore, the volume of the greenhouse is about 3 greenhouse? 32.2 ft .

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 26. The base of the pyramid is a regular octagon with SOLUTION: sides of 2 feet. A central angle of the octagon is Volume of the solid given = Volume of the small or 45°, so the angle formed in the triangle cone + Volume of the large cone below is 22.5°.

Use a trigonometric ratio to find the apothem a.

ANSWER: 471.2 in3

The height of this pyramid is 5 feet.

27. SOLUTION:

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft ANSWER: Find the volume of each solid. Round to the 3 nearest tenth. 3190.6 m

26. 28. SOLUTION: Volume of the solid given = Volume of the small SOLUTION: cone + Volume of the large cone

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her ANSWER: backyard. To buy a heating unit for the space, she 471.2 in3 needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

27. SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

ANSWER: 3 3190.6 m The volume of the ceiling is

28.

SOLUTION: The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on ANSWER: rock formations.For a project, she is making a clay 7698.5 cm3 model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of 29. HEATING Sam is building an art studio in her each pyramid will be 1 by 1.5 inches, and the total backyard. To buy a heating unit for the space, she height will be 4 inches. needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction Determine the volume of the model. Explain why with good insulation, there should be 2 BTUs per knowing the volume is helpful in this situation. cubic foot. What size unit does Sam need to SOLUTION: purchase? The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base It tells Marta how much clay is needed to make the is model.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

The volume of the ceiling is 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled.

The total volume is therefore 5000 + 1666.67 = SOLUTION: 3 Find the volume of the original cone. Then alter the 6666.67 ft . Two BTU's are needed for every cubic values. foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of a. Double h. two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches.

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION: The volume is doubled. The volume of a pyramid is , where B is the b. Double r. area of the base and h is the height of the pyramid.

2 It tells Marta how much clay is needed to make the The volume is multiplied by 2 or 4. model. c. Double r and h. ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. volume is multiplied by 23 or 8. a. The height is doubled. b. The radius is doubled. ANSWER: c. Both the radius and the height are doubled. a. The volume is doubled. 2 SOLUTION: b. The volume is multiplied by 2 or 4. Find the volume of the original cone. Then alter the c. The volume is multiplied by 23 or 8. values. Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the a. Double h. area of the base and h is the height of the pyramid. Let s be the side length of the base.

The volume is doubled.

b. Double r.

The side length of the base is 15 cm.

2 ANSWER: The volume is multiplied by 2 or 4. 15 cm c. Double r and h. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the volume is multiplied by 23 or 8. height of the cone, and r is the radius of the base. ANSWER: Since the diameter is 8 centimeters, the radius is 4 centimeters. a. The volume is doubled.

2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The diameter is 2(7) or 14 inches. Let s be the side length of the base. ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The side length of the base is 15 cm.

ANSWER: 15 cm The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the 33. The volume of a cone is 196π cubic inches and the radius r. height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. So, the radius is about 3.8 millimeters. Since the diameter is 8 centimeters, the radius is 4 centimeters. Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where The diameter is 2(7) or 14 inches. r is the radius of the base and h is the height of the ANSWER: cone. 14 in.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this

The lateral area of a cone is , where r is problem, you will investigate rectangular pyramids. the radius and is the slant height of the cone. a. GEOMETRIC Draw two pyramids with Replace L with 71.6 and with 6, then solve for the different bases that have a height of 10 centimeters

radius r. and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the

two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION:

So, the radius is about 3.8 millimeters. a. Use rectangular bases and pick values that multiply to make 24. Use the Pythagorean Theorem to find the height of the cone. Sample answer:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the c. If the base area is multiplied by 5, the volume is volume of the pyramid. multiplied by 5. If the height is multiplied by 5, the 12-5 Volumes of Pyramids and Cones volume is multiplied by 5. If both the base area and SOLUTION: the height are multiplied by 5, the volume is multiplied a. Use rectangular bases and pick values that by 5 · 5 or 25. multiply to make 24.

Sample answer:

ANSWER: a. Sample answer:

b. The volumes are the same. The volume of a eSolutions Manual - Powered by Cognero pyramid equals one third times the base area timesPage 15 the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

ANSWER: a. Sample answer:

b. The volumes are the same. The volume of a or 10h cubic units. pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are ANSWER: equal and their heights are equal, then their volumes Sometimes; the statement is true if the base area of are equal. the cone is 3 times as great as the base area of the c. If the base area is multiplied by 5, the volume is prism. For example, if the base of the prism has an multiplied by 5. If the height is multiplied by 5, the area of 10 square units, then its volume is 10h cubic volume is multiplied by 5. If both the base area and units. So, the cone must have a base area of 30 the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. square units so that its volume is or 10h

36. CCSS ARGUMENTS Determine whether the cubic units. following statement is sometimes, always, or never 37. ERROR ANALYSIS Alexandra and Cornelio are true. Justify your reasoning. calculating the volume of the cone below. Is either of

The volume of a cone with radius r and height h them correct? Explain your answer. equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

Alexandra incorrectly used the slant height.

ANSWER: The volumes will only be equal when the radius of Cornelio; Alexandra incorrectly used the slant height. the cone is equal to or when . 38. REASONING A cone has a volume of 568 cubic Therefore, the statement is true sometimes if the centimeters. What is the volume of a cylinder that base area of the cone is 3 times as great as the base has the same radius and height as the cone? Explain area of the prism. For example, if the base of the your reasoning. prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base SOLUTION: 3 area of 30 square units so that its volume is 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone or 10h cubic units. is V = Bh. The volume of a cylinder is three times ANSWER: as much as the volume of a cone with the same radius and height. Sometimes; the statement is true if the base area of the cone is 3 times as great as the base area of the ANSWER: prism. For example, if the base of the prism has an 3 area of 10 square units, then its volume is 10h cubic 1704 cm ; The volume of a cylinder is three times as units. So, the cone must have a base area of 30 much as the volume of a cone with the same radius and height. square units so that its volume is or 10h 39. OPEN ENDED Give an example of a pyramid and cubic units. a prism that have the same base and the same volume. Explain your reasoning. 37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of SOLUTION: them correct? Explain your answer. The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

SOLUTION:

The slant height is used for surface area, but the Set the base areas of the prism and pyramid, and height is used for volume. For this cone, the slant make the height of the pyramid equal to 3 times the height of 13 is provided, and we need to calculate the height of the prism.

height before we can calculate the volume.

Sample answer:

A square pyramid with a base area of 16 and a

Alexandra incorrectly used the slant height. height of 12, a prism with a square base area of 16 ANSWER: and a height of 4. Cornelio; Alexandra incorrectly used the slant height. ANSWER: Sample answer: A square pyramid with a base area 38. REASONING A cone has a volume of 568 cubic of 16 and a height of 12, a prism with a square base centimeters. What is the volume of a cylinder that area of 16 and a height of 4; if a pyramid and prism has the same radius and height as the cone? Explain have the same base, then in order to have the same your reasoning. volume, the height of the pyramid must be 3 times as SOLUTION: great as the height of the prism. 3 1704 cm ; The formula for the volume of a cylinder 40. WRITING IN MATH Compare and contrast is V= Bh, while the formula for the volume of a cone finding volumes of pyramids and cones with finding is V = Bh. The volume of a cylinder is three times volumes of prisms and cylinders. as much as the volume of a cone with the same SOLUTION: radius and height. To find the volume of each solid, you must know the ANSWER: area of the base and the height. The volume of a 3 pyramid is one third the volume of a prism that has 1704 cm ; The volume of a cylinder is three times as the same height and base area. The volume of a much as the volume of a cone with the same radius cone is one third the volume of a cylinder that has the and height. same height and base area.

39. OPEN ENDED Give an example of a pyramid and ANSWER: a prism that have the same base and the same To find the volume of each solid, you must know the volume. Explain your reasoning. area of the base and the height. The volume of a SOLUTION: pyramid is one third the volume of a prism that has The formula for volume of a prism is V = Bh and the the same height and base area. The volume of a formula for the volume of a pyramid is one-third of cone is one third the volume of a cylinder that has the

that. So, if a pyramid and prism have the same base, same height and base area. then in order to have the same volume, the height of 41. A conical sand toy has the dimensions as shown the pyramid must be 3 times as great as the height of below. How many cubic centimeters of sand will it

the prism. hold when it is filled to the top?

A 12π B 15π Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the C height of the prism.

D Sample answer: A square pyramid with a base area of 16 and a SOLUTION: height of 12, a prism with a square base area of 16 Use the Pythagorean Theorem to find the radius r of and a height of 4. the cone. ANSWER: Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

pyramid is one third the volume of a prism that has So, the radius of the cone is 3 centimeters. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the The volume of a circular cone is , where same height and base area. r is the radius of the base and h is the height of the ANSWER: cone. To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it Therefore, the correct choice is A. hold when it is filled to the top? ANSWER: A

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is A 12π 6 feet by 8 feet. If the tent holds 88 cubic feet of air, B 15π how tall is the tent’s center pole?

C SOLUTION:

The volume of a pyramid is , where B is the D area of the base and h is the height of the pyramid. SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the

cone. F

J Therefore, the correct choice is A. SOLUTION: ANSWER: Possible outcomes: {6 red, 4 blue, 5 orange, 10 A green}

42. SHORT RESPONSE Brooke is buying a tent that Number of possible outcomes : 25

is in the shape of a rectangular pyramid. The base is Favorable outcomes: {5 orange}

6 feet by 8 feet. If the tent holds 88 cubic feet of air, Number of favorable outcomes: 5 how tall is the tent’s center pole?

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the correct choice is F.

ANSWER: F

44. SAT/ACT For all

A –8 B x – 4 ANSWER: C 5.5 ft D PROBABILITY 43. A spinner has sections colored E red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange? SOLUTION:

So, the correct choice is E.

F ANSWER: E G Find the volume of each prism. H

J 45. SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 SOLUTION: green} The volume of a prism is , where B is the Number of possible outcomes : 25 area of the base and h is the height of the prism. The Favorable outcomes: {5 orange} base of this prism is a rectangle with a length of 14 Number of favorable outcomes: 5 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the correct choice is F. 3 Therefore, the volume of the prism is 1008 in . ANSWER: ANSWER: F 1008 in3 44. SAT/ACT For all

A –8 B x – 4 C 46. D SOLUTION: E The volume of a prism is , where B is the area of the base and h is the height of the prism. The SOLUTION: base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the correct choice is E.

ANSWER: E

Find the volume of each prism.

45.

SOLUTION: So, the height of the triangle is 12 feet. Find the area The volume of a prism is , where B is the of the triangle. area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet. 3 Therefore, the volume of the prism is 1008 in .

ANSWER: 1008 in3

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 1140 ft 46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will 47. bisect the base. Use the Pythagorean Theorem to find the height of the triangle. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

So, the height of the triangle is 12 feet. Find the area ANSWER: of the triangle. 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write So, the area of the base B is 60 ft2. the exact answer and the answer rounded to the nearest square foot. The height h of the prism is 19 feet. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is

3 , where is r is the radius and l is the slant height Therefore, the volume of the prism is 1140 ft . of the cone. ANSWER: Find the slant height of the conical top. 3 1140 ft

47. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the Find the slant height of the conical bottom. prism is 102.3 meters. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain The formula for finding the surface area of a cylinder after harvest. The cone at the bottom of the bin is , where is r is the radius and h is the slant allows the grain to be emptied more easily. Use the height of the cylinder. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface SOLUTION: area of the conical bottom. To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

ANSWER: Find the slant height of the conical top.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) Find the slant height of the conical bottom. = 50 The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

ANSWER: 30.4 cm2

50. The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant SOLUTION: height of the cylinder. Area of the shaded region = Area of the circle – Area of the hexagon

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom. A regular hexagon has 6 sides, so the measure of the

interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: 30.4 cm2 Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

ANSWER: 2 54.4 in A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem. 51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the Now find the area of the hexagon. height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

ANSWER: 2 54.4 in Now use the area of the isosceles triangles to find the area of the equilateral triangle.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle. So, the area of the equilateral triangle is about 67.3 square feet. Find the area of the circle with a radius of 3.6 feet. Subtract to find the area of shaded region.

So, the area of the circle is about 40.7 square feet. Therefore, the area of the shaded region is about Next, find the area of the equilateral triangle. 26.6 ft2.

ANSWER: 2 26.6 ft

The equilateral triangle can be divided into three 52. isosceles triangles with central angles of or SOLUTION: 120°, so the angle in the triangle created by the The area of the shaded region is the area of the height of the isosceles triangle is 60° and the base is circumscribed circle minus the area of the equilateral half the length b of the side of the equilateral triangle. triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Use a trigonometric ratio to find the value of b.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the Now use the area of the isosceles triangles to find isosceles triangle is 60° and the base is half the the area of the equilateral triangle. length of the side of the equilateral triangle as shown below.

So, the area of the equilateral triangle is about 67.3 square feet. Use trigonometric ratios to find the values of b and h. Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 The area of the equilateral triangle will equal three 26.6 ft times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A

52. (equilateral triangle) + A(inscribed circle) SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral Therefore, the area of the shaded region is about triangle and h represent the radius of the inscribed 2 circle. 168.2 mm .

ANSWER: 2 168.2 mm

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

The volume of a pyramid is , where B is the

Find the volume of each pyramid. area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the pyramid is 10 inches.

1. SOLUTION:

3 75 in

2. ANSWER: SOLUTION: 3 75 in The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

2. SOLUTION:

3 132 cm 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length ANSWER: 3 of 8 meters and a width of 4.5 meters. The height of 62.4 m the pyramid is 5.2 meters. 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 ANSWER: meters. The height of the pyramid is 14 meters. 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: The base of this pyramid is a square with sides of 8 298.7 m3 meters. The height of the pyramid is 14 meters. Find the volume of each cone. Round to the nearest tenth.

5. ANSWER: SOLUTION: 298.7 m3 The volume of a circular cone is , or Find the volume of each cone. Round to the nearest tenth. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches. 5.

SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the

height of the cone, and r is the radius of the base. ANSWER: Since the diameter of this cone is 7 inches, the radius 3 is or 3.5 inches. The height of the cone is 4 inches. 51.3 in

6. SOLUTION: ANSWER: 3 51.3 in

Use trigonometry to find the radius r.

SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

Use trigonometry to find the radius r.

The volume of a circular cone is , or ANSWER: , where B is the area of the base, h is the 168.1 cm3 height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the ANSWER: 3 height of the cone, and r is the radius of the base. 168.1 cm The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters

SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. ANSWER: 3 The radius of this cone is 1.6 millimeters and the 28.1 mm height is 10.5 millimeters. 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

ANSWER: 3 28.1 mm Use the Pythagorean Theorem to find the height h of 8. a cone with a slant height of 25 meters and a radius the cone. Then find its volume. of 15 meters

SOLUTION:

So, the height of the cone is 20 meters. Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

ANSWER: 4712.4 m3 So, the height of the cone is 20 meters. 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION: ANSWER: The volume of a circular cone is , where B 4712.4 m3 is the area of the base and h is the height of the 9. MUSEUMS The sky dome of the National Corvette cone. Museum in Bowling Green, Kentucky, is a conical For this cone, the area of the base is 15,400 square building. If the height is 100 feet and the area of the feet and the height is 100 feet. base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. ANSWER: For this cone, the area of the base is 15,400 square 3 feet and the height is 100 feet. 513,333.3 ft CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

ANSWER:

3 10. 513,333.3 ft SOLUTION: CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if The volume of a pyramid is , where B is the necessary. area of the base and h is the height of the pyramid.

10. SOLUTION: ANSWER: The volume of a pyramid is , where B is the 605 in3 area of the base and h is the height of the pyramid.

11. SOLUTION: ANSWER: 605 in3 The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

11. SOLUTION: ANSWER: The volume of a pyramid is , where B is the 3 105.8 mm area of the base and h is the height of the pyramid.

12. SOLUTION:

ANSWER: The volume of a pyramid is , where B is the 3 105.8 mm area of the base and h is the height of the pyramid.

12. SOLUTION: ANSWER: The volume of a pyramid is , where B is the 482.1 m3 area of the base and h is the height of the pyramid.

13. SOLUTION: ANSWER: 482.1 m3 The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle. 13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

The apothem is . ANSWER: 3 233.8 cm 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER:

3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION: ANSWER: The volume of a pyramid is , where B is the 1376.7 ft3 area of the base and h is the height of the pyramid. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

The volume of a pyramid is , where B is the ANSWER: 3 area of the base and h is the height of the pyramid. 35.6 cm 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a ANSWER: of the right triangle and then find the area of the 3 35.6 cm triangle.

The length of the other leg of the right triangle is 6 cm.

2 So, the area of the base B is 24 cm . The length of the other leg of the right triangle is 6 cm. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. Therefore, the height of the triangular pyramid is 18 cm. ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth.

Therefore, the height of the triangular pyramid is 18 cm. 17.

ANSWER: SOLUTION: 18 cm The volume of a circular cone is , Find the volume of each cone. Round to the where r is the radius of the base and h is the height nearest tenth. of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 17. inches.

SOLUTION: The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the 3 Therefore, the volume of the cone is about 235.6 in . radius is or 5 inches. The height of the cone is 9 inches. ANSWER: 235.6 in3

18. 3 Therefore, the volume of the cone is about 235.6 in . SOLUTION: ANSWER: The volume of a circular cone is , where 235.6 in3 r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 134.8 3 cone. The radius of this cone is 4.2 centimeters and cm . the height is 7.3 centimeters. ANSWER: 134.8 cm3

Therefore, the volume of the cone is about 134.8 19. 3 cm . SOLUTION: ANSWER: 134.8 cm3

19. Use a trigonometric ratio to find the height h of the cone. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. Use a trigonometric ratio to find the height h of the cone.

The volume of a circular cone is , where Therefore, the volume of the cone is about 1473.1 3 r is the radius of the base and h is the height of the cm . cone. The radius of this cone is 8 centimeters. ANSWER: 1473.1 cm3

20. Therefore, the volume of the cone is about 1473.1 3 cm . SOLUTION:

ANSWER: 1473.1 cm3

Use trigonometric ratios to find the height h and the radius r of the cone. 20. SOLUTION:

The volume of a circular cone is , where

Use trigonometric ratios to find the height h and the r is the radius of the base and h is the height of the radius r of the cone. cone.

The volume of a circular cone is , where 3 r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 2.8 ft . cone. ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where 3 Therefore, the volume of the cone is about 2.8 ft . r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, ANSWER: the radius is or 8 inches. 3 2.8 ft 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where

r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 1072.3 cone. Since the diameter of this cone is 16 inches, 3 in . the radius is or 8 inches. ANSWER: 1072.3 in3

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant Therefore, the volume of the cone is about 1072.3 height of 5.6 centimeters. Use the Pythagorean 3 in . Theorem to find the height h of the cone.

ANSWER: 1072.3 in3

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3

ANSWER: The volume of a circular cone is , where 5.8 cm3 r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a centimeters, the radius is or 4 centimeters. The paper cone that is 14 centimeters high and has a base height of the cone is 14 centimeters. diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the

cone. Since the diameter of the cone is 8 3 Therefore, the paper cone will hold about 234.6 cm centimeters, the radius is or 4 centimeters. The of roasted peanuts. height of the cone is 14 centimeters. ANSWER: 234.6 cm3

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this

3 pyramid. Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. SOLUTION:

ANSWER: The volume of a pyramid is , where B is the 234.6 cm3 area of the base and h is the height of the pyramid.

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION: ANSWER: The volume of a pyramid is , where B is the 3 42,000,000 ft area of the base and h is the height of the pyramid. 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base SOLUTION: has side lengths of 2 feet. What is the volume of the greenhouse? The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

SOLUTION:

below is 22.5°.

The height of this pyramid is 5 feet.

Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet. Therefore, the volume of the greenhouse is about 3 32.2 ft . ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

Therefore, the volume of the greenhouse is about 32.2 ft3. 26. ANSWER: SOLUTION: 3 32.2 ft Volume of the solid given = Volume of the small cone + Volume of the large cone Find the volume of each solid. Round to the nearest tenth.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 471.2 in3

27.

ANSWER: SOLUTION: 471.2 in3

27. ANSWER: SOLUTION: 3 3190.6 m

28. SOLUTION: ANSWER: 3 3190.6 m

28. SOLUTION: ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction SOLUTION: with good insulation, there should be 2 BTUs per The building can be broken down into the rectangular cubic foot. What size unit does Sam need to base and the pyramid ceiling. The volume of the base purchase? is

The volume of the ceiling is SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is The volume of the ceiling is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay The total volume is therefore 5000 + 1666.67 = model of a crystal with a shape that is a composite of 3 6666.67 ft . Two BTU's are needed for every cubic two congruent rectangular pyramids. The base of foot, so the size of the heating unit Sam should buy is each pyramid will be 1 by 1.5 inches, and the total 6666.67 × 2 = 13,333 BTUs. height will be 4 inches.

Determine the volume of the model. Explain why ANSWER: knowing the volume is helpful in this situation. 13,333 BTUs SOLUTION: 30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay The volume of a pyramid is , where B is the model of a crystal with a shape that is a composite of area of the base and h is the height of the pyramid. two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches.

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION: It tells Marta how much clay is needed to make the The volume of a pyramid is , where B is the model. area of the base and h is the height of the pyramid. ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. It tells Marta how much clay is needed to make the a. The height is doubled.

model. b. The radius is doubled. ANSWER: c. Both the radius and the height are doubled. 2 in3; It tells Marta how much clay is needed to SOLUTION: make the model. Find the volume of the original cone. Then alter the values. 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled.

SOLUTION: a. Double h. Find the volume of the original cone. Then alter the values.

The volume is doubled.

b. Double r. a. Double h.

2 The volume is doubled. The volume is multiplied by 2 or 4.

b. Double r. c. Double r and h.

2 3 The volume is multiplied by 2 or 4. volume is multiplied by 2 or 8.

ANSWER: c. Double r and h. a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic 3 centimeters and a height of 11.5 centimeters. Find volume is multiplied by 2 or 8. the side length of the base. ANSWER: SOLUTION: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. The volume of a pyramid is , where B is the c. The volume is multiplied by 23 or 8. area of the base and h is the height of the pyramid. Let s be the side length of the base. Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or The side length of the base is 15 cm. , where B is the area of the base, h is the ANSWER: 15 cm height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter?

SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The diameter is 2(7) or 14 inches.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The diameter is 2(7) or 14 inches.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? The lateral area of a cone is , where r is SOLUTION: the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

The lateral area of a cone is , where r is So, the radius is about 3.8 millimeters. the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the Use the Pythagorean Theorem to find the height of radius r. the cone.

So, the radius is about 3.8 millimeters.

So, the height of the cone is about 4.64 millimeters. Use the Pythagorean Theorem to find the height of the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where Therefore, the volume of the cone is about 70.2 3 r is the radius of the base and h is the height of the mm . cone. ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters Therefore, the volume of the cone is about 70.2 and a base area of 24 square centimeters. 3 b. VERBAL What is true about the volumes of the mm . two pyramids that you drew? Explain. ANSWER: c. ANALYTICAL Explain how multiplying the base 3 area and/or the height of the pyramid by 5 affects the 70.2 mm volume of the pyramid. 35. MULTIPLE REPRESENTATIONS In this SOLUTION: problem, you will investigate rectangular pyramids. a. Use rectangular bases and pick values that a. GEOMETRIC Draw two pyramids with multiply to make 24. different bases that have a height of 10 centimeters and a base area of 24 square centimeters. Sample answer: b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

ANSWER: a. Sample answer:

b. The volumes are the same. The volume of a 36. CCSS ARGUMENTS Determine whether the pyramid equals one third times the base area times following statement is sometimes, always, or never the height. So, if the base areas of two pyramids are true. Justify your reasoning. equal and their heights are equal, then their volumes The volume of a cone with radius r and height h are equal. equals the volume of a prism with height h. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the SOLUTION: volume is multiplied by 5. If both the base area and The volume of a cone with a radius r and height h is the height are multiplied by 5, the volume is multiplied . The volume of a prism with a height of by 5 · 5 or 25. h is where B is the area of the base of the prism. Set the volumes equal. 36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

or 10h cubic units. The volumes will only be equal when the radius of ANSWER: the cone is equal to or when . Sometimes; the statement is true if the base area of Therefore, the statement is true sometimes if the the cone is 3 times as great as the base area of the base area of the cone is 3 times as great as the base prism. For example, if the base of the prism has an area of the prism. For example, if the base of the area of 10 square units, then its volume is 10h cubic prism has an area of 10 square units, then its volume units. So, the cone must have a base area of 30 is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is square units so that its volume is or 10h

or 10h cubic units. cubic units. 37. ERROR ANALYSIS Alexandra and Cornelio are ANSWER: calculating the volume of the cone below. Is either of Sometimes; the statement is true if the base area of them correct? Explain your answer. the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an eSolutionsarea Manualof 10 square- Powered units,by Cognero then its volume is 10h cubic Page 16 units. So, the cone must have a base area of 30

square units so that its volume is or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer. SOLUTION: The slant height is used for surface area, but the height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume.

Alexandra incorrectly used the slant height.

ANSWER:

Cornelio; Alexandra incorrectly used the slant height. SOLUTION: The slant height is used for surface area, but the 38. REASONING A cone has a volume of 568 cubic height is used for volume. For this cone, the slant centimeters. What is the volume of a cylinder that height of 13 is provided, and we need to calculate the has the same radius and height as the cone? Explain height before we can calculate the volume. your reasoning.

SOLUTION:

3 Alexandra incorrectly used the slant height. 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone ANSWER: is V = Bh. The volume of a cylinder is three times Cornelio; Alexandra incorrectly used the slant height. as much as the volume of a cone with the same

38. REASONING A cone has a volume of 568 cubic radius and height. centimeters. What is the volume of a cylinder that ANSWER: has the same radius and height as the cone? Explain 3 your reasoning. 1704 cm ; The volume of a cylinder is three times as much as the volume of a cone with the same radius SOLUTION: and height. 3 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same is V = Bh. The volume of a cylinder is three times volume. Explain your reasoning. as much as the volume of a cone with the same radius and height. SOLUTION: The formula for volume of a prism is V = Bh and the ANSWER: formula for the volume of a pyramid is one-third of 1704 cm3; The volume of a cylinder is three times as that. So, if a pyramid and prism have the same base, much as the volume of a cone with the same radius then in order to have the same volume, the height of and height. the pyramid must be 3 times as great as the height of the prism. 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, Set the base areas of the prism and pyramid, and then in order to have the same volume, the height of make the height of the pyramid equal to 3 times the the pyramid must be 3 times as great as the height of height of the prism. the prism. Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

ANSWER: Sample answer: A square pyramid with a base area Set the base areas of the prism and pyramid, and of 16 and a height of 12, a prism with a square base make the height of the pyramid equal to 3 times the area of 16 and a height of 4; if a pyramid and prism height of the prism. have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as Sample answer: great as the height of the prism. A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 40. WRITING IN MATH Compare and contrast and a height of 4. finding volumes of pyramids and cones with finding volumes of prisms and cylinders. ANSWER: SOLUTION: Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base To find the volume of each solid, you must know the area of 16 and a height of 4; if a pyramid and prism area of the base and the height. The volume of a have the same base, then in order to have the same pyramid is one third the volume of a prism that has volume, the height of the pyramid must be 3 times as the same height and base area. The volume of a great as the height of the prism. cone is one third the volume of a cylinder that has the same height and base area. 40. WRITING IN MATH Compare and contrast ANSWER: finding volumes of pyramids and cones with finding volumes of prisms and cylinders. To find the volume of each solid, you must know the area of the base and the height. The volume of a SOLUTION: pyramid is one third the volume of a prism that has To find the volume of each solid, you must know the the same height and base area. The volume of a area of the base and the height. The volume of a cone is one third the volume of a cylinder that has the pyramid is one third the volume of a prism that has same height and base area. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the 41. A conical sand toy has the dimensions as shown same height and base area. below. How many cubic centimeters of sand will it hold when it is filled to the top? ANSWER: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a A 12π cone is one third the volume of a cylinder that has the same height and base area. B 15π

C 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top? D

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

A 12π B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

So, the radius of the cone is 3 centimeters.

The volume of a circular cone is , where Therefore, the correct choice is A. r is the radius of the base and h is the height of the cone. ANSWER: A

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole?

SOLUTION: Therefore, the correct choice is A. The volume of a pyramid is , where B is the ANSWER: area of the base and h is the height of the pyramid. A

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

ANSWER:

5.5 ft F 43. PROBABILITY A spinner has sections colored

red, blue, orange, and green. The table below shows G the results of several spins. What is the experimental probability of the spinner landing on orange? H

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 F green} Number of possible outcomes : 25 G Favorable outcomes: {5 orange} Number of favorable outcomes: 5

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 So, the correct choice is F. Favorable outcomes: {5 orange} Number of favorable outcomes: 5 ANSWER: F

44. SAT/ACT For all

A –8 B x – 4 C

So, the correct choice is F. D ANSWER: E F SOLUTION: 44. SAT/ACT For all

A –8 B x – 4 C So, the correct choice is E. D ANSWER: E E

SOLUTION: Find the volume of each prism.

45. So, the correct choice is E. SOLUTION: ANSWER: The volume of a prism is , where B is the E area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 Find the volume of each prism. inches and a width of 12 inches. The height h of the prism is 6 inches.

45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The 3 base of this prism is a rectangle with a length of 14 Therefore, the volume of the prism is 1008 in . inches and a width of 12 inches. The height h of the ANSWER: prism is 6 inches. 3 1008 in

46. 3 Therefore, the volume of the prism is 1008 in . SOLUTION: ANSWER: The volume of a prism is , where B is the area of the base and h is the height of the prism. The 1008 in3 base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

46. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the height of the triangle is 12 feet. Find the area of the triangle.

2 So, the area of the base B is 60 ft . So, the height of the triangle is 12 feet. Find the area of the triangle. The height h of the prism is 19 feet.

Therefore, the volume of the prism is 1140 ft3.

2 So, the area of the base B is 60 ft . ANSWER: 3 1140 ft The height h of the prism is 19 feet.

47. Therefore, the volume of the prism is 1140 ft3. SOLUTION: ANSWER: The volume of a prism is , where B is the 3 1140 ft area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

47.

SOLUTION: The volume of a prism is , where B is the Therefore, the volume of the prism is about 426,437.6 area of the base and h is the height of the prism. The m3. base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the ANSWER: prism is 102.3 meters. 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the

dimensions in the diagram to find the entire surface Therefore, the volume of the prism is about 426,437.6 area of the bin with a conical top and bottom. Write 3 m . the exact answer and the answer rounded to the nearest square foot. ANSWER: Refer to the photo on Page 863. 3 426,437.6 m SOLUTION: 48. FARMING The picture shows a combination To find the entire surface area of the bin, find the hopper cone and bin used by farmers to store grain surface area of the conical top and bottom and find after harvest. The cone at the bottom of the bin the surface area of the cylinder and add them. allows the grain to be emptied more easily. Use the The formula for finding the surface area of a cone is dimensions in the diagram to find the entire surface , where is r is the radius and l is the slant height area of the bin with a conical top and bottom. Write of the cone. the exact answer and the answer rounded to the nearest square foot. Find the slant height of the conical top. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

ANSWER: Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom. Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: ANSWER: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50 Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – ANSWER: Area of the circle 30.4 cm2 Area of the rectangle = 10(5) = 50

50. SOLUTION: Area of the shaded region = Area of the circle – ANSWER: Area of the hexagon 30.4 cm2

A regular hexagon has 6 sides, so the measure of the 50. SOLUTION: interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the Area of the shaded region = Area of the circle – length of the apothem. Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Now find the area of the hexagon.

ANSWER: 2 54.4 in

51.

SOLUTION: The shaded area is the area of the equilateral triangle ANSWER: less the area of the inscribed circle.

2 54.4 in Find the area of the circle with a radius of 3.6 feet.

51. So, the area of the circle is about 40.7 square feet. SOLUTION: The shaded area is the area of the equilateral triangle Next, find the area of the equilateral triangle. less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet. The equilateral triangle can be divided into three isosceles triangles with central angles of or Next, find the area of the equilateral triangle. 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

The equilateral triangle can be divided into three Use a trigonometric ratio to find the value of b. isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

So, the area of the equilateral triangle is about 67.3

square feet.

Now use the area of the isosceles triangles to find

the area of the equilateral triangle. Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 26.6 ft2. So, the area of the equilateral triangle is about 67.3 square feet. ANSWER: 2 26.6 ft Subtract to find the area of shaded region.

52. Therefore, the area of the shaded region is about SOLUTION: 2 26.6 ft . The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral ANSWER: triangle plus the area of the inscribed circle. Let b 2 26.6 ft represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b Divide the equilateral triangle into three isosceles represent the length of each side of the equilateral triangles with central angles of or 120°. The triangle and h represent the radius of the inscribed angle in the triangle formed by the height.h of the circle. isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the Use trigonometric ratios to find the values of b and isosceles triangle is 60° and the base is half the h. length of the side of the equilateral triangle as shown below.

Use trigonometric ratios to find the values of b and h. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle) ANSWER: 2 168.2 mm

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

1. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs ANSWER: of 9 inches and 5 inches and the height of the 3 132 cm pyramid is 10 inches. 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

ANSWER: 3 2. 62.4 m SOLUTION: 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with The volume of a pyramid is , where B is the sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 area of the base and h is the height of the pyramid. centimeters. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

ANSWER: 298.7 m3 ANSWER: 132 cm3 Find the volume of each cone. Round to the nearest tenth. 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

The volume of a pyramid is , where B is the 5. area of the base and h is the height of the pyramid. SOLUTION: The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of The volume of a circular cone is , or the pyramid is 5.2 meters. , where B is the area of the base, h is the

height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths ANSWER: SOLUTION: 3 51.3 in The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

6. SOLUTION:

ANSWER: 3 298.7 m Use trigonometry to find the radius r.

Find the volume of each cone. Round to the nearest tenth.

The volume of a circular cone is , or

, where B is the area of the base, h is the

5. height of the cone, and r is the radius of the base. SOLUTION: The height of the cone is 11.5 centimeters.

The volume of a circular cone is , or

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or ANSWER: , where B is the area of the base, h is the 3 51.3 in height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

6. SOLUTION:

ANSWER: 3 28.1 mm

Use trigonometry to find the radius r. 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

ANSWER: 3 168.1 cm So, the height of the cone is 20 meters. 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the ANSWER: height of the cone, and r is the radius of the base. 3 The radius of this cone is 1.6 millimeters and the 4712.4 m height is 10.5 millimeters. 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

ANSWER: The volume of a circular cone is , where B 3 28.1 mm is the area of the base and h is the height of the cone. 8. a cone with a slant height of 25 meters and a radius of 15 meters For this cone, the area of the base is 15,400 square feet and the height is 100 feet. SOLUTION:

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

So, the height of the cone is 20 meters.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 4712.4 m3

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet. 11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if ANSWER: necessary. 3 105.8 mm

10. 12. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

ANSWER: 605 in3 ANSWER: 482.1 m3

11. 13. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 3 105.8 mm

12. SOLUTION:

The volume of a pyramid is , where B is the The apothem is . area of the base and h is the height of the pyramid.

ANSWER: 3 482.1 m ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION: 13. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle. ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The apothem is .

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet

SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 3 35.6 cm 1376.7 ft3 16. A triangular pyramid with a right triangle base with a 15. a triangular pyramid with a height of 4.8 centimeters leg 8 centimeters and hypotenuse 10 centimeters has and a right triangle base with a leg 5 centimeters and a volume of 144 cubic centimeters. Find the height. hypotenuse 10.2 centimeters SOLUTION: SOLUTION: The base of the pyramid is a right triangle with a leg Find the height of the right triangle. of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

The length of the other leg of the right triangle is 6 cm.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Therefore, the height of the triangular pyramid is 18 Use the Pythagorean Theorem to find the other leg a cm. of the right triangle and then find the area of the ANSWER: triangle. 18 cm Find the volume of each cone. Round to the nearest tenth.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The length of the other leg of the right triangle is 6 Since the diameter of this cone is 10 inches, the cm. radius is or 5 inches. The height of the cone is 9

inches.

So, the area of the base B is 24 cm2.

3 Replace V with 144 and B with 24 in the formula for Therefore, the volume of the cone is about 235.6 in . the volume of a pyramid and solve for the height h. ANSWER: 235.6 in3

18.

Therefore, the height of the triangular pyramid is 18 SOLUTION: cm. The volume of a circular cone is , where ANSWER: r is the radius of the base and h is the height of the 18 cm cone. The radius of this cone is 4.2 centimeters and

Find the volume of each cone. Round to the the height is 7.3 centimeters. nearest tenth.

17. SOLUTION: Therefore, the volume of the cone is about 134.8 3 The volume of a circular cone is , cm .

where r is the radius of the base and h is the height ANSWER:

of the cone. 3 134.8 cm Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

19. SOLUTION:

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 3 235.6 in Use a trigonometric ratio to find the height h of the cone.

18. SOLUTION: The volume of a circular cone is , where The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the volume of the cone is about 1473.1 3 cm . Therefore, the volume of the cone is about 134.8 3 ANSWER: cm . 1473.1 cm3 ANSWER: 134.8 cm3

20. SOLUTION: 19. SOLUTION:

Use trigonometric ratios to find the height h and the radius r of the cone.

Use a trigonometric ratio to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the The volume of a circular cone is , where cone.

r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

3 Therefore, the volume of the cone is about 2.8 ft . Therefore, the volume of the cone is about 1473.1 ANSWER: 3 cm . 3 2.8 ft ANSWER: 21. an oblique cone with a diameter of 16 inches and an 3 1473.1 cm altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the 20. cone. Since the diameter of this cone is 16 inches, SOLUTION: the radius is or 8 inches.

Use trigonometric ratios to find the height h and the radius r of the cone. Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the The cone has a radius r of 1 centimeter and a slant cone. height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

3 Therefore, the volume of the cone is about 5.8 cm . ANSWER: 5.8 cm3

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a Therefore, the volume of the cone is about 1072.3 paper cone that is 14 centimeters high and has a base 3 in . diameter of 8 centimeters? Round to the nearest tenth. ANSWER: SOLUTION: 1072.3 in3 The volume of a circular cone is , where 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 SOLUTION: The cone has a radius r of 1 centimeter and a slant centimeters, the radius is or 4 centimeters. The height of 5.6 centimeters. Use the Pythagorean height of the cone is 14 centimeters. Theorem to find the height h of the cone.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest ANSWER: 3 tenth. 42,000,000 ft SOLUTION: 25. GARDENING The greenhouse is a regular The volume of a circular cone is , where octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the r is the radius of the base and h is the height of the greenhouse? cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

SOLUTION:

The volume of a pyramid is , where B is the 3 Therefore, the paper cone will hold about 234.6 cm area of the base and h is the height of the pyramid. of roasted peanuts. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is ANSWER: or 45°, so the angle formed in the triangle 3 234.6 cm below is 22.5°.

The volume of a pyramid is , where B is the Use a trigonometric ratio to find the apothem a. area of the base and h is the height of the pyramid.

The height of this pyramid is 5 feet.

ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the SOLUTION: nearest tenth.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is 26. or 45°, so the angle formed in the triangle SOLUTION: below is 22.5°. Volume of the solid given = Volume of the small cone + Volume of the large cone

Use a trigonometric ratio to find the apothem a.

ANSWER: The height of this pyramid is 5 feet. 3 471.2 in

27. SOLUTION:

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth. ANSWER: 3 3190.6 m

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone 28.

SOLUTION:

ANSWER: 3 ANSWER: 7698.5 cm 3 471.2 in 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase? 27. SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is ANSWER: 3 3190.6 m

The volume of the ceiling is

28. SOLUTION:

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: ANSWER: 13,333 BTUs 3 7698.5 cm 30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay 29. HEATING Sam is building an art studio in her model of a crystal with a shape that is a composite of backyard. To buy a heating unit for the space, she two congruent rectangular pyramids. The base of needs to determine the BTUs (British Thermal Units) each pyramid will be 1 by 1.5 inches, and the total required to heat the building. For new construction height will be 4 inches. with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to Determine the volume of the model. Explain why purchase? knowing the volume is helpful in this situation.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

It tells Marta how much clay is needed to make the model.

ANSWER: 2 in3; It tells Marta how much clay is needed to The volume of the ceiling is make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled.

b. The total volume is therefore 5000 + 1666.67 = The radius is doubled. 3 c. Both the radius and the height are doubled. 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is SOLUTION: 6666.67 × 2 = 13,333 BTUs. Find the volume of the original cone. Then alter the values. ANSWER: 13,333 BTUs

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The volume is doubled.

b. Double r.

It tells Marta how much clay is needed to make the model. 2 ANSWER: The volume is multiplied by 2 or 4.

2 in3; It tells Marta how much clay is needed to c. Double r and h. make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. 3 c. Both the radius and the height are doubled. volume is multiplied by 2 or 8. SOLUTION: ANSWER: Find the volume of the original cone. Then alter the a. The volume is doubled. values. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base.

a. Double h. SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The volume is doubled.

b. Double r.

2 The volume is multiplied by 2 or 4. The side length of the base is 15 cm.

c. Double r and h. ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or volume is multiplied by 23 or 8.

ANSWER: , where B is the area of the base, h is the a. The volume is doubled. height of the cone, and r is the radius of the base. 2 b. The volume is multiplied by 2 or 4. Since the diameter is 8 centimeters, the radius is 4 3 centimeters. c. The volume is multiplied by 2 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The diameter is 2(7) or 14 inches. ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the The lateral area of a cone is , where r is height is 12 inches. What is the diameter? the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the SOLUTION: radius r.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4

centimeters.

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters. The diameter is 2(7) or 14 inches. ANSWER: The volume of a circular cone is , where 14 in. r is the radius of the base and h is the height of the 34. The lateral area of a cone is 71.6 square millimeters cone. and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3 The lateral area of a cone is , where r is the radius and is the slant height of the cone. 35. MULTIPLE REPRESENTATIONS In this Replace L with 71.6 and with 6, then solve for the problem, you will investigate rectangular pyramids. radius r. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. So, the radius is about 3.8 millimeters. SOLUTION: Use the Pythagorean Theorem to find the height of a. Use rectangular bases and pick values that the cone. multiply to make 24.

Sample answer:

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

b. The volumes are the same. The volume of a Therefore, the volume of the cone is about 70.2 pyramid equals one third times the base area times 3 the height. So, if the base areas of two pyramids are mm . equal and their heights are equal, then their volumes

ANSWER: are equal. 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: c. If the base area is multiplied by 5, the volume is a. Use rectangular bases and pick values that multiplied by 5. If the height is multiplied by 5, the multiply to make 24. volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. Sample answer:

ANSWER: a. Sample answer:

a. Sample answer:

The volumes will only be equal when the radius of the cone is equal to or when . Therefore, the statement is true sometimes if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume b. The volumes are the same. The volume of a is 10h cubic units. So, the cone must have a base pyramid equals one third times the base area times area of 30 square units so that its volume is the height. So, if the base areas of two pyramids are or 10h cubic units. equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is ANSWER: multiplied by 5. If the height is multiplied by 5, the Sometimes; the statement is true if the base area of volume is multiplied by 5. If both the base area and the cone is 3 times as great as the base area of the the height are multiplied by 5, the volume is multiplied prism. For example, if the base of the prism has an by 5 · 5 or 25. area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never square units so that its volume is or 10h true. Justify your reasoning. cubic units. The volume of a cone with radius r and height h equals the volume of a prism with height h. 37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of SOLUTION: them correct? Explain your answer. The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

The volumes will only be equal when the radius of Alexandra incorrectly used the slant height. the cone is equal to or when . ANSWER: Therefore, the statement is true sometimes if the Cornelio; Alexandra incorrectly used the slant height. base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the 38. REASONING A cone has a volume of 568 cubic prism has an area of 10 square units, then its volume centimeters. What is the volume of a cylinder that is 10h cubic units. So, the cone must have a base has the same radius and height as the cone? Explain area of 30 square units so that its volume is your reasoning.

or 10h cubic units. SOLUTION: 3 1704 cm ; The formula for the volume of a cylinder ANSWER: is V= Bh, while the formula for the volume of a cone Sometimes; the statement is true if the base area of is V = Bh. The volume of a cylinder is three times the cone is 3 times as great as the base area of the as much as the volume of a cone with the same prism. For example, if the base of the prism has an radius and height. area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 ANSWER: 3 square units so that its volume is or 10h 1704 cm ; The volume of a cylinder is three times as 12-5 Volumes of Pyramids and Cones much as the volume of a cone with the same radius cubic units. and height.

37. ERROR ANALYSIS Alexandra and Cornelio are 39. OPEN ENDED Give an example of a pyramid and calculating the volume of the cone below. Is either of a prism that have the same base and the same them correct? Explain your answer. volume. Explain your reasoning. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

SOLUTION: The slant height is used for surface area, but the height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume. Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism. Alexandra incorrectly used the slant height. Sample answer: ANSWER: A square pyramid with a base area of 16 and a Cornelio; Alexandra incorrectly used the slant height. height of 12, a prism with a square base area of 16 and a height of 4. 38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that ANSWER: has the same radius and height as the cone? Explain Sample answer: A square pyramid with a base area your reasoning. of 16 and a height of 12, a prism with a square base area of 16 and a height of 4; if a pyramid and prism SOLUTION: have the same base, then in order to have the same 3 1704 cm ; The formula for the volume of a cylinder volume, the height of the pyramid must be 3 times as is V= Bh, while the formula for the volume of a cone great as the height of the prism. is V = Bh. The volume of a cylinder is three times 40. WRITING IN MATH Compare and contrast as much as the volume of a cone with the same finding volumes of pyramids and cones with finding radius and height. volumes of prisms and cylinders. ANSWER: SOLUTION: 1704 cm3; The volume of a cylinder is three times as To find the volume of each solid, you must know the much as the volume of a cone with the same radius area of the base and the height. The volume of a and height. pyramid is one third the volume of a prism that has the same height and base area. The volume of a 39. OPEN ENDED Give an example of a pyramid and cone is one third the volume of a cylinder that has the a prism that have the same base and the same same height and base area. volume. Explain your reasoning. ANSWER: SOLUTION: To find the volume of each solid, you must know the The formula for volume of a prism is V = Bh and the area of the base and the height. The volume of a formula for the volume of a pyramid is one-third of pyramid is one third the volume of a prism that has that. So, if a pyramid and prism have the same base, the same height and base area. The volume of a then in order to have the same volume, the height of cone is one third the volume of a cylinder that has the the pyramid must be 3 times as great as the height of same height and base area. eSolutionsthe prism.Manual - Powered by Cognero Page 17 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the A 12π height of the prism. B 15π

C Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 D and a height of 4. SOLUTION: ANSWER: Use the Pythagorean Theorem to find the radius r of Sample answer: A square pyramid with a base area the cone. of 16 and a height of 12, a prism with a square base area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders.

SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area. So, the radius of the cone is 3 centimeters. ANSWER: The volume of a circular cone is , where To find the volume of each solid, you must know the r is the radius of the base and h is the height of the area of the base and the height. The volume of a cone. pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

Therefore, the correct choice is A.

ANSWER: A A 12π B 15π 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is C 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole?

D SOLUTION:

SOLUTION: The volume of a pyramid is , where B is the Use the Pythagorean Theorem to find the radius r of area of the base and h is the height of the pyramid.

the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows So, the radius of the cone is 3 centimeters. the results of several spins. What is the experimental probability of the spinner landing on orange? The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

H Therefore, the correct choice is A.

ANSWER: J A SOLUTION: 42. SHORT RESPONSE Brooke is buying a tent that Possible outcomes: {6 red, 4 blue, 5 orange, 10 is in the shape of a rectangular pyramid. The base is green} 6 feet by 8 feet. If the tent holds 88 cubic feet of air, Number of possible outcomes : 25 how tall is the tent’s center pole? Favorable outcomes: {5 orange} SOLUTION: Number of favorable outcomes: 5

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

So, the correct choice is F.

ANSWER: F

44. SAT/ACT For all ANSWER: A –8 5.5 ft B x – 4

43. PROBABILITY A spinner has sections colored C red, blue, orange, and green. The table below shows D the results of several spins. What is the experimental probability of the spinner landing on orange? E

SOLUTION:

F So, the correct choice is E. G ANSWER:

H E Find the volume of each prism. J

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} 45. Number of possible outcomes : 25 SOLUTION: Favorable outcomes: {5 orange} Number of favorable outcomes: 5 The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the correct choice is F.

ANSWER: 3 F Therefore, the volume of the prism is 1008 in .

ANSWER: 44. SAT/ACT For all 1008 in3 A –8 B x – 4 C D E 46. SOLUTION: SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to

So, the correct choice is E. find the height of the triangle.

ANSWER: E Find the volume of each prism.

45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 So, the height of the triangle is 12 feet. Find the area inches and a width of 12 inches. The height h of the of the triangle. prism is 6 inches.

So, the area of the base B is 60 ft2. 3 Therefore, the volume of the prism is 1008 in . The height h of the prism is 19 feet. ANSWER: 3 1008 in

Therefore, the volume of the prism is 1140 ft3.

46. ANSWER: 3 SOLUTION: 1140 ft The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle. 47. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 So, the height of the triangle is 12 feet. Find the area 3 of the triangle. m .

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the So, the area of the base B is 60 ft2. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write The height h of the prism is 19 feet. the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find Therefore, the volume of the prism is 1140 ft3. the surface area of the cylinder and add them. The formula for finding the surface area of a cone is ANSWER: , where is r is the radius and l is the slant height 3 1140 ft of the cone.

Find the slant height of the conical top.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface The formula for finding the surface area of a cylinder area of the bin with a conical top and bottom. Write is , where is r is the radius and h is the slant the exact answer and the answer rounded to the height of the cylinder. nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the Surface area of the bin = Surface area of the surface area of the conical top and bottom and find cylinder + Surface area of the conical top + surface the surface area of the cylinder and add them. area of the conical bottom. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION:

Area of the shaded region = Area of the rectangle – Find the slant height of the conical bottom. Area of the circle The height of the conical bottom is 28 – (5 + 12 + 2) Area of the rectangle = 10(5) or 9 ft. = 50

ANSWER: 2 30.4 cm

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder. 50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the ANSWER: length of the apothem.

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49.

SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: 30.4 cm2

Now find the area of the hexagon.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the ANSWER: 2 interior angle is . The apothem bisects the 54.4 in angle, so we use 30° when using trig to find the length of the apothem.

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three

Now find the area of the hexagon. isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

ANSWER: 2 54.4 in

Now use the area of the isosceles triangles to find the area of the equilateral triangle. 51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft

The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the 52. height of the isosceles triangle is 60° and the base is SOLUTION: half the length b of the side of the equilateral triangle. The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed

circle.

Use a trigonometric ratio to find the value of b.

Divide the equilateral triangle into three isosceles Now use the area of the isosceles triangles to find triangles with central angles of or 120°. The the area of the equilateral triangle. angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region. Use trigonometric ratios to find the values of b and

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region. 52. SOLUTION: A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle) The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle. Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

Find the volume of each pyramid.

1. ANSWER: 3 SOLUTION: 75 in

SOLUTION:

2. ANSWER: SOLUTION: 132 cm3 The volume of a pyramid is , where B is the 3. a rectangular pyramid with a height of 5.2 meters area of the base and h is the height of the pyramid. and a base 8 meters by 4.5 meters The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 SOLUTION: centimeters. The height of the pyramid is 12 centimeters. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 132 cm3 ANSWER: 3 3. a rectangular pyramid with a height of 5.2 meters 62.4 m and a base 8 meters by 4.5 meters 4. a square pyramid with a height of 14 meters and a SOLUTION: base with 8-meter side lengths

The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a rectangle with a length area of the base and h is the height of the pyramid. of 8 meters and a width of 4.5 meters. The height of The base of this pyramid is a square with sides of 8 the pyramid is 5.2 meters. meters. The height of the pyramid is 14 meters.

ANSWER: 3 ANSWER: 62.4 m 298.7 m3 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths Find the volume of each cone. Round to the nearest tenth. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 5. meters. The height of the pyramid is 14 meters. SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius ANSWER: is or 3.5 inches. The height of the cone is 4 inches. 298.7 m3

Find the volume of each cone. Round to the nearest tenth.

ANSWER: 5. 3 51.3 in SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. 6. Since the diameter of this cone is 7 inches, the radius SOLUTION: is or 3.5 inches. The height of the cone is 4 inches.

Use trigonometry to find the radius r.

ANSWER: 3 51.3 in The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

6. SOLUTION:

ANSWER: 168.1 cm3 Use trigonometry to find the radius r. 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: The volume of a circular cone is , or The volume of a circular cone is , or , where B is the area of the base, h is the , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and ANSWER: 3 a radius of 1.6 millimeters 28.1 mm

SOLUTION: 8. a cone with a slant height of 25 meters and a radius of 15 meters The volume of a circular cone is , or SOLUTION: , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius So, the height of the cone is 20 meters. of 15 meters SOLUTION:

ANSWER: Use the Pythagorean Theorem to find the height h of 3 the cone. Then find its volume. 4712.4 m

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION: So, the height of the cone is 20 meters. The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 4712.4 m3

9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of ANSWER: 3 air that the heating and cooling systems would have 513,333.3 ft to accommodate. Round to the nearest tenth. CCSS SENSE-MAKING Find the volume of SOLUTION: each pyramid. Round to the nearest tenth if necessary. The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet. 10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. ANSWER: 605 in3

10. SOLUTION: 11. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: 605 in3 ANSWER: 3 105.8 mm

11. SOLUTION: 12. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 105.8 mm ANSWER: 482.1 m3

12. SOLUTION: 13. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the

base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle. ANSWER: 482.1 m3

13. SOLUTION:

The volume of a pyramid is , where B is the

base and h is the height of the pyramid. The apothem is .

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The apothem is . The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 1376.7 ft3 3 233.8 cm 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and 14. a pentagonal pyramid with a base area of 590 square hypotenuse 10.2 centimeters feet and an altitude of 7 feet SOLUTION: SOLUTION: Find the height of the right triangle. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION:

The base of the pyramid is a right triangle with a leg The volume of a pyramid is , where B is the of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a area of the base and h is the height of the pyramid. of the right triangle and then find the area of the

triangle.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. The length of the other leg of the right triangle is 6 Use the Pythagorean Theorem to find the other leg a cm. of the right triangle and then find the area of the triangle.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The length of the other leg of the right triangle is 6 cm.

Therefore, the height of the triangular pyramid is 18 cm.

ANSWER: 18 cm

So, the area of the base B is 24 cm2. Find the volume of each cone. Round to the nearest tenth. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the height of the triangular pyramid is 18 Since the diameter of this cone is 10 inches, the cm. radius is or 5 inches. The height of the cone is 9 ANSWER: inches. 18 cm

Find the volume of each cone. Round to the nearest tenth.

3 17. Therefore, the volume of the cone is about 235.6 in .

SOLUTION: ANSWER: The volume of a circular cone is , 235.6 in3 where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 18. inches. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3

Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 18. 134.8 cm3 SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. 19.

SOLUTION:

Therefore, the volume of the cone is about 134.8 3 cm . Use a trigonometric ratio to find the height h of the ANSWER: cone. 134.8 cm3

The volume of a circular cone is , where 19. r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. SOLUTION:

Therefore, the volume of the cone is about 1473.1 Use a trigonometric ratio to find the height h of the 3 cone. cm . ANSWER: 1473.1 cm3

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. 20. SOLUTION:

Therefore, the volume of the cone is about 1473.1 Use trigonometric ratios to find the height h and the 3 cm . radius r of the cone.

ANSWER: 1473.1 cm3

The volume of a circular cone is , where 20. r is the radius of the base and h is the height of the SOLUTION: cone.

Use trigonometric ratios to find the height h and the radius r of the cone. 3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 3 Therefore, the volume of the cone is about 1072.3 2.8 ft 3 in . 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches ANSWER: 3 SOLUTION: 1072.3 in

The volume of a circular cone is , where 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, SOLUTION: The cone has a radius r of 1 centimeter and a slant the radius is or 8 inches. height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3

23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a 3 paper cone that is 14 centimeters high and has a base Therefore, the paper cone will hold about 234.6 cm diameter of 8 centimeters? Round to the nearest of roasted peanuts. tenth. ANSWER: SOLUTION: 234.6 cm3 The volume of a circular cone is , where 24. CCSS MODELING The Pyramid Arena in r is the radius of the base and h is the height of the Memphis, Tennessee, is the third largest pyramid in cone. Since the diameter of the cone is 8 the world. It is approximately 350 feet tall, and its centimeters, the radius is or 4 centimeters. The square base is 600 feet wide. Find the volume of this height of the cone is 14 centimeters. pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 3 234.6 cm ANSWER: 3 24. CCSS MODELING The Pyramid Arena in 42,000,000 ft Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its 25. GARDENING The greenhouse is a regular square base is 600 feet wide. Find the volume of this octagonal pyramid with a height of 5 feet. The base pyramid. has side lengths of 2 feet. What is the volume of the greenhouse? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: The base of the pyramid is a regular octagon with 3 42,000,000 ft sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle GARDENING 25. The greenhouse is a regular below is 22.5°. octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Use a trigonometric ratio to find the apothem a.

SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with The height of this pyramid is 5 feet. sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Therefore, the volume of the greenhouse is about Use a trigonometric ratio to find the apothem a. 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth. The height of this pyramid is 5 feet.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

ANSWER: 3 26. 471.2 in SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

27. SOLUTION:

ANSWER: 3 ANSWER: 471.2 in 3 3190.6 m

27. 28. SOLUTION: SOLUTION:

ANSWER: 3 3190.6 m ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction 28. with good insulation, there should be 2 BTUs per SOLUTION: cubic foot. What size unit does Sam need to purchase?

SOLUTION: The building can be broken down into the rectangular ANSWER: base and the pyramid ceiling. The volume of the base 7698.5 cm3 is

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to The volume of the ceiling is purchase?

The total volume is therefore 5000 + 1666.67 = 3 SOLUTION: 6666.67 ft . Two BTU's are needed for every cubic The building can be broken down into the rectangular foot, so the size of the heating unit Sam should buy is base and the pyramid ceiling. The volume of the base 6666.67 × 2 = 13,333 BTUs. is

ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on

rock formations.For a project, she is making a clay The volume of the ceiling is model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches.

Determine the volume of the model. Explain why knowing the volume is helpful in this situation.

SOLUTION: The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic The volume of a pyramid is , where B is the foot, so the size of the heating unit Sam should buy is area of the base and h is the height of the pyramid. 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of It tells Marta how much clay is needed to make the two congruent rectangular pyramids. The base of model. each pyramid will be 1 by 1.5 inches, and the total ANSWER: height will be 4 inches. 3 2 in ; It tells Marta how much clay is needed to Determine the volume of the model. Explain why make the model. knowing the volume is helpful in this situation. 31. CHANGING DIMENSIONS A cone has a radius SOLUTION: of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the The volume of a pyramid is , where B is the cone. area of the base and h is the height of the pyramid. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

It tells Marta how much clay is needed to make the model.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius a. Double h. of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled.

SOLUTION: The volume is doubled. Find the volume of the original cone. Then alter the values. b. Double r.

2 The volume is multiplied by 2 or 4. a. Double h.

c. Double r and h.

The volume is doubled.

3 b. Double r. volume is multiplied by 2 or 8. ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth 2 The volume is multiplied by 2 or 4. if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find c. Double r and h. the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

3 volume is multiplied by 2 or 8. ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base.

SOLUTION: The side length of the base is 15 cm.

The volume of a pyramid is , where B is the ANSWER: area of the base and h is the height of the pyramid. 15 cm Let s be the side length of the base. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or The diameter is 2(7) or 14 inches.

, where B is the area of the base, h is the ANSWER: 14 in. height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 34. The lateral area of a cone is 71.6 square millimeters centimeters. and the slant height is 6 millimeters. What is the volume of the cone?

SOLUTION:

The lateral area of a cone is , where r is The diameter is 2(7) or 14 inches. the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the ANSWER: radius r. 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r. So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the

cone.

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3 So, the height of the cone is about 4.64 millimeters. 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. The volume of a circular cone is , where a. GEOMETRIC Draw two pyramids with r is the radius of the base and h is the height of the different bases that have a height of 10 centimeters cone. and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that Therefore, the volume of the cone is about 70.2 multiply to make 24. 3 mm . Sample answer: ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer: b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal.

c. If the base area is multiplied by 5, the volume is b. The volumes are the same. The volume of a multiplied by 5. If the height is multiplied by 5, the pyramid equals one third times the base area times volume is multiplied by 5. If both the base area and the height. So, if the base areas of two pyramids are the height are multiplied by 5, the volume is multiplied equal and their heights are equal, then their volumes by 5 · 5 or 25. are equal.

ANSWER: a. Sample answer:

ANSWER: a. Sample answer: b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

b. The volumes are the same. The volume of a . The volume of a prism with a height of pyramid equals one third times the base area times h is where B is the area of the base of the the height. So, if the base areas of two pyramids are prism. Set the volumes equal. equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h

equals the volume of a prism with height h. SOLUTION: The volumes will only be equal when the radius of The volume of a cone with a radius r and height h is the cone is equal to or when . . The volume of a prism with a height of Therefore, the statement is true sometimes if the h is where B is the area of the base of the base area of the cone is 3 times as great as the base prism. Set the volumes equal. area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

square units so that its volume is or 10h The volumes will only be equal when the radius of cubic units. the cone is equal to or when . Therefore, the statement is true sometimes if the 37. ERROR ANALYSIS Alexandra and Cornelio are base area of the cone is 3 times as great as the base calculating the volume of the cone below. Is either of area of the prism. For example, if the base of the them correct? Explain your answer. prism has an area of 10 square units, then its volume is 10h cubic units. So, the cone must have a base area of 30 square units so that its volume is

or 10h cubic units.

ANSWER: Sometimes; the statement is true if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic SOLUTION: units. So, the cone must have a base area of 30 The slant height is used for surface area, but the square units so that its volume is or 10h height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the cubic units. height before we can calculate the volume.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of Alexandra incorrectly used the slant height. them correct? Explain your answer. ANSWER: Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. SOLUTION: 3 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone SOLUTION: The slant height is used for surface area, but the is V = Bh. The volume of a cylinder is three times height is used for volume. For this cone, the slant as much as the volume of a cone with the same height of 13 is provided, and we need to calculate the radius and height. height before we can calculate the volume. ANSWER: 1704 cm3; The volume of a cylinder is three times as Alexandra incorrectly used the slant height. much as the volume of a cone with the same radius and height. ANSWER: Cornelio; Alexandra incorrectly used the slant height. 39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same 38. REASONING A cone has a volume of 568 cubic volume. Explain your reasoning. centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain SOLUTION: your reasoning. The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of SOLUTION: that. So, if a pyramid and prism have the same base, 3 1704 cm ; The formula for the volume of a cylinder then in order to have the same volume, the height of is V= Bh, while the formula for the volume of a cone the pyramid must be 3 times as great as the height of the prism. is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

ANSWER: 1704 cm3; The volume of a cylinder is three times as much as the volume of a cone with the same radius and height. Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the 39. OPEN ENDED Give an example of a pyramid and height of the prism. a prism that have the same base and the same volume. Explain your reasoning. Sample answer: SOLUTION: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 The formula for volume of a prism is V = Bh and the and a height of 4. formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, ANSWER: then in order to have the same volume, the height of Sample answer: A square pyramid with a base area the pyramid must be 3 times as great as the height of of 16 and a height of 12, a prism with a square base the prism. area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the SOLUTION: height of the prism. To find the volume of each solid, you must know the area of the base and the height. The volume of a Sample answer: pyramid is one third the volume of a prism that has A square pyramid with a base area of 16 and a the same height and base area. The volume of a height of 12, a prism with a square base area of 16 cone is one third the volume of a cylinder that has the and a height of 4. same height and base area.

ANSWER: ANSWER: Sample answer: A square pyramid with a base area To find the volume of each solid, you must know the of 16 and a height of 12, a prism with a square base area of the base and the height. The volume of a area of 16 and a height of 4; if a pyramid and prism pyramid is one third the volume of a prism that has have the same base, then in order to have the same the same height and base area. The volume of a 12-5volume, Volumes the of height Pyramids of the andpyramid Cones must be 3 times as cone is one third the volume of a cylinder that has the great as the height of the prism. same height and base area.

40. WRITING IN MATH Compare and contrast 41. A conical sand toy has the dimensions as shown finding volumes of pyramids and cones with finding below. How many cubic centimeters of sand will it volumes of prisms and cylinders. hold when it is filled to the top? SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a A 12π cone is one third the volume of a cylinder that has the B 15π same height and base area. C ANSWER:

To find the volume of each solid, you must know the D area of the base and the height. The volume of a pyramid is one third the volume of a prism that has SOLUTION: the same height and base area. The volume of a Use the Pythagorean Theorem to find the radius r of cone is one third the volume of a cylinder that has the the cone. same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

A 12π B 15π

D So, the radius of the cone is 3 centimeters. SOLUTION: Use the Pythagorean Theorem to find the radius r of The volume of a circular cone is , where the cone. r is the radius of the base and h is the height of the cone.

Therefore, the correct choice is A. ANSWER: A

42. SHORT RESPONSE Brooke is buying a tent that eSolutions Manual - Powered by Cognero is in the shape of a rectangular pyramid. The basePage is18 So, the radius of the cone is 3 centimeters. 6 feet by 8 feet. If the tent holds 88 cubic feet of air, The volume of a circular cone is , where how tall is the tent’s center pole? r is the radius of the base and h is the height of the SOLUTION: cone. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

Therefore, the correct choice is A.

ANSWER: A

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is ANSWER: 6 feet by 8 feet. If the tent holds 88 cubic feet of air, 5.5 ft how tall is the tent’s center pole? 43. PROBABILITY A spinner has sections colored SOLUTION: red, blue, orange, and green. The table below shows the results of several spins. What is the experimental The volume of a pyramid is , where B is the probability of the spinner landing on orange? area of the base and h is the height of the pyramid.

ANSWER: J 5.5 ft SOLUTION: 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows Possible outcomes: {6 red, 4 blue, 5 orange, 10

the results of several spins. What is the experimental green}

probability of the spinner landing on orange? Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

G So, the correct choice is F. H ANSWER:

J F

SOLUTION: 44. SAT/ACT For all Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} A –8 Number of possible outcomes : 25 B x – 4 Favorable outcomes: {5 orange} C Number of favorable outcomes: 5 D E

SOLUTION:

So, the correct choice is F.

ANSWER: So, the correct choice is E. F ANSWER: E 44. SAT/ACT For all Find the volume of each prism. A –8 B x – 4 C

D 45. E SOLUTION: The volume of a prism is , where B is the SOLUTION: area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the correct choice is E.

ANSWER: E

Find the volume of each prism. 3 Therefore, the volume of the prism is 1008 in .

ANSWER: 1008 in3 45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the 46.

prism is 6 inches. SOLUTION:

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 1008 in3

46. SOLUTION:

The volume of a prism is , where B is the So, the height of the triangle is 12 feet. Find the area area of the base and h is the height of the prism. The of the triangle. base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

Therefore, the volume of the prism is 1140 ft3. So, the height of the triangle is 12 feet. Find the area of the triangle. ANSWER: 3 1140 ft

So, the area of the base B is 60 ft2. 47. SOLUTION: The height h of the prism is 19 feet. The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 1140 ft Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m 47. 48. FARMING The picture shows a combination SOLUTION: hopper cone and bin used by farmers to store grain The volume of a prism is , where B is the after harvest. The cone at the bottom of the bin area of the base and h is the height of the prism. The allows the grain to be emptied more easily. Use the base of this prism is a rectangle with a length of 79.4 dimensions in the diagram to find the entire surface meters and a width of 52.5 meters. The height of the area of the bin with a conical top and bottom. Write prism is 102.3 meters. the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find Therefore, the volume of the prism is about 426,437.6 the surface area of the cylinder and add them. The formula for finding the surface area of a cone is m3. , where is r is the radius and l is the slant height ANSWER: of the cone.

3 426,437.6 m Find the slant height of the conical top.

48. FARMING The picture shows a combination

hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find Find the slant height of the conical bottom. the surface area of the cylinder and add them. The height of the conical bottom is 28 – (5 + 12 + 2) The formula for finding the surface area of a cone is or 9 ft. , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2)

or 9 ft. Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

ANSWER:

The formula for finding the surface area of a cylinder Find the area of each shaded region. Polygons is , where is r is the radius and h is the slant in 50 - 52 are regular. height of the cylinder.

49.

Surface area of the bin = Surface area of the SOLUTION: cylinder + Surface area of the conical top + surface Area of the shaded region = Area of the rectangle – area of the conical bottom. Area of the circle Area of the rectangle = 10(5) = 50

ANSWER:

ANSWER: Find the area of each shaded region. Polygons 30.4 cm2 in 50 - 52 are regular.

49. 50. SOLUTION: Area of the shaded region = Area of the rectangle – SOLUTION: Area of the circle Area of the shaded region = Area of the circle – Area of the rectangle = 10(5) Area of the hexagon = 50

A regular hexagon has 6 sides, so the measure of the

ANSWER: interior angle is . The apothem bisects the 2 angle, so we use 30° when using trig to find the 30.4 cm length of the apothem.

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the

interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Now find the area of the hexagon.

ANSWER: 2 54.4 in

Now find the area of the hexagon. 51.

SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet. ANSWER: 2 Next, find the area of the equilateral triangle. 54.4 in

51. SOLUTION: The shaded area is the area of the equilateral triangle The equilateral triangle can be divided into three less the area of the inscribed circle. isosceles triangles with central angles of or

120°, so the angle in the triangle created by the

Find the area of the circle with a radius of 3.6 feet. height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle. Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find The equilateral triangle can be divided into three the area of the equilateral triangle. isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region. Use a trigonometric ratio to find the value of b.

Therefore, the area of the shaded region is about 2 26.6 ft . Now use the area of the isosceles triangles to find the area of the equilateral triangle. ANSWER: 2 26.6 ft

52. So, the area of the equilateral triangle is about 67.3 SOLUTION: square feet. The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral Subtract to find the area of shaded region. triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed circle.

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the

52. length of the side of the equilateral triangle as shown SOLUTION: below. The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral triangle and h represent the radius of the inscribed

circle. Use trigonometric ratios to find the values of b and h.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below. The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Use trigonometric ratios to find the values of b and h.

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

Find the volume of each pyramid.

ANSWER: 3 75 in

1. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. 2. The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the SOLUTION: pyramid is 10 inches. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

ANSWER: 3 75 in

ANSWER: 132 cm3 2. 3. a rectangular pyramid with a height of 5.2 meters SOLUTION: and a base 8 meters by 4.5 meters

The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a regular pentagon with area of the base and h is the height of the pyramid. sides of 4.4 centimeters and an apothem of 3 The base of this pyramid is a rectangle with a length centimeters. The height of the pyramid is 12 of 8 meters and a width of 4.5 meters. The height of centimeters. the pyramid is 5.2 meters.

ANSWER: 3 62.4 m ANSWER: 132 cm3 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths 3. a rectangular pyramid with a height of 5.2 meters SOLUTION: and a base 8 meters by 4.5 meters SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The volume of a pyramid is , where B is the The base of this pyramid is a square with sides of 8 area of the base and h is the height of the pyramid. meters. The height of the pyramid is 14 meters. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 298.7 m3

ANSWER: Find the volume of each cone. Round to the 3 62.4 m nearest tenth.

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION: 5. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 The volume of a circular cone is , or meters. The height of the pyramid is 14 meters. , where B is the area of the base, h is the

height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth. ANSWER: 3 51.3 in

5. SOLUTION:

Use trigonometry to find the radius r.

The volume of a circular cone is , or

ANSWER: , where B is the area of the base, h is the 3 51.3 in height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

6. SOLUTION: ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters

Use trigonometry to find the radius r. SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the The volume of a circular cone is , or height of the cone, and r is the radius of the base. , where B is the area of the base, h is the The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters. height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

ANSWER: ANSWER: 3 28.1 mm 168.1 cm3 8. a cone with a slant height of 25 meters and a radius 7. an oblique cone with a height of 10.5 millimeters and of 15 meters a radius of 1.6 millimeters SOLUTION: SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Use the Pythagorean Theorem to find the height h of The radius of this cone is 1.6 millimeters and the the cone. Then find its volume. height is 10.5 millimeters.

ANSWER: So, the height of the cone is 20 meters.

3 28.1 mm 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

ANSWER: 4712.4 m3

9. MUSEUMS The sky dome of the National Corvette Use the Pythagorean Theorem to find the height h of Museum in Bowling Green, Kentucky, is a conical the cone. Then find its volume. building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the So, the height of the cone is 20 meters. cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

The volume of a circular cone is , where B

is the area of the base and h is the height of the 10. cone. For this cone, the area of the base is 15,400 square SOLUTION: feet and the height is 100 feet. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 513,333.3 ft ANSWER: CCSS SENSE-MAKING Find the volume of 605 in3 each pyramid. Round to the nearest tenth if necessary.

ANSWER: ANSWER: 3 105.8 mm 605 in3

12. 11. SOLUTION:

SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

area of the base and h is the height of the pyramid.

ANSWER: 3 ANSWER: 482.1 m 3 105.8 mm

ANSWER: 482.1 m3

13. SOLUTION: The apothem is .

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

The apothem is .

ANSWER: 1376.7 ft3

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 1376.7 ft3

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. The length of the other leg of the right triangle is 6 cm. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

The length of the other leg of the right triangle is 6 Therefore, the height of the triangular pyramid is 18 cm. cm.

ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for 17. the volume of a pyramid and solve for the height h. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

ANSWER: 3 17. 235.6 in SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. 18.

Since the diameter of this cone is 10 inches, the SOLUTION: radius is or 5 inches. The height of the cone is 9 The volume of a circular cone is , where inches. r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 3 Therefore, the volume of the cone is about 134.8 235.6 in 3 cm .

ANSWER: 134.8 cm3

18. SOLUTION:

The volume of a circular cone is , where 19. r is the radius of the base and h is the height of the SOLUTION: cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Use a trigonometric ratio to find the height h of the

cone. Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 134.8 cm3 The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

19. SOLUTION:

Therefore, the volume of the cone is about 1473.1 3 cm .

ANSWER:

Use a trigonometric ratio to find the height h of the 1473.1 cm3 cone.

20. The volume of a circular cone is , where SOLUTION: r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Use trigonometric ratios to find the height h and the radius r of the cone.

Therefore, the volume of the cone is about 1473.1 3 cm .

ANSWER: 1473.1 cm3 The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

20. SOLUTION:

3 Therefore, the volume of the cone is about 2.8 ft . Use trigonometric ratios to find the height h and the ANSWER: radius r of the cone. 3 2.8 ft 21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches

SOLUTION:

The volume of a circular cone is , where The volume of a circular cone is , where r is the radius of the base and h is the height of the r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, cone. the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 3 3 Therefore, the volume of the cone is about 2.8 ft . in .

ANSWER: ANSWER: 3 2.8 ft 1072.3 in3

21. an oblique cone with a diameter of 16 inches and an 22. a right cone with a slant height of 5.6 centimeters altitude of 16 inches and a radius of 1 centimeter SOLUTION: SOLUTION: The volume of a circular cone is , where The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean r is the radius of the base and h is the height of the Theorem to find the height h of the cone. cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 3 1072.3 in 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone. 3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8

centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters.

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER:

3 3 5.8 cm Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a ANSWER: paper cone that is 14 centimeters high and has a base 3 diameter of 8 centimeters? Round to the nearest 234.6 cm tenth. 24. CCSS MODELING The Pyramid Arena in SOLUTION: Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its The volume of a circular cone is , where square base is 600 feet wide. Find the volume of this

r is the radius of the base and h is the height of the pyramid. cone. Since the diameter of the cone is 8 SOLUTION:

centimeters, the radius is or 4 centimeters. The The volume of a pyramid is , where B is the height of the cone is 14 centimeters. area of the base and h is the height of the pyramid.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts. ANSWER: 3 42,000,000 ft ANSWER: 234.6 cm3 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base 24. CCSS MODELING The Pyramid Arena in has side lengths of 2 feet. What is the volume of the Memphis, Tennessee, is the third largest pyramid in greenhouse? the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. SOLUTION:

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

Use a trigonometric ratio to find the apothem a.

SOLUTION: The height of this pyramid is 5 feet. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 Use a trigonometric ratio to find the apothem a. 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

The height of this pyramid is 5 feet. 26.

SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth. ANSWER: 471.2 in3

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone 27. SOLUTION:

ANSWER: 3 3190.6 m ANSWER: 471.2 in3

28. SOLUTION: 27. SOLUTION:

ANSWER: 7698.5 cm3 ANSWER: 3 29. HEATING Sam is building an art studio in her 3190.6 m backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

28. SOLUTION:

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

ANSWER: 7698.5 cm3

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

SOLUTION: The building can be broken down into the rectangular ANSWER: base and the pyramid ceiling. The volume of the base 13,333 BTUs is 30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches. The volume of the ceiling is Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the

ANSWER: 13,333 BTUs It tells Marta how much clay is needed to make the model. 30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay ANSWER: model of a crystal with a shape that is a composite of 2 in3; It tells Marta how much clay is needed to two congruent rectangular pyramids. The base of make the model. each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches. 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Determine the volume of the model. Explain why Describe how each change affects the volume of the knowing the volume is helpful in this situation. cone. a. The height is doubled. SOLUTION: b. The radius is doubled. The volume of a pyramid is , where B is the c. Both the radius and the height are doubled. area of the base and h is the height of the pyramid. SOLUTION: Find the volume of the original cone. Then alter the values.

It tells Marta how much clay is needed to make the model.

ANSWER: a. Double h. 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone.

a. The height is doubled. The volume is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. b. Double r. SOLUTION: Find the volume of the original cone. Then alter the values.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

a. Double h.

volume is multiplied by 23 or 8.

The volume is doubled. ANSWER: a. The volume is doubled. b. Double r. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

c. Double r and h. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

volume is multiplied by 23 or 8.

ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic The side length of the base is 15 cm. centimeters and a height of 11.5 centimeters. Find the side length of the base. ANSWER: 15 cm SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? The diameter is 2(7) or 14 inches.

SOLUTION: ANSWER: 14 in. The volume of a circular cone is , or 34. The lateral area of a cone is 71.6 square millimeters , where B is the area of the base, h is the and the slant height is 6 millimeters. What is the volume of the cone? height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 SOLUTION: centimeters.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r. The diameter is 2(7) or 14 inches.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? So, the radius is about 3.8 millimeters.

SOLUTION: Use the Pythagorean Theorem to find the height of the cone.

r is the radius of the base and h is the height of the cone.

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone. Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this

problem, you will investigate rectangular pyramids. So, the height of the cone is about 4.64 millimeters. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters

and a base area of 24 square centimeters. The volume of a circular cone is , where b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. r is the radius of the base and h is the height of the c. ANALYTICAL Explain how multiplying the base cone. area and/or the height of the pyramid by 5 affects the volume of the pyramid.

SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer: Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid.

SOLUTION: a. Use rectangular bases and pick values that b. The volumes are the same. The volume of a multiply to make 24. pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are Sample answer: equal and their heights are equal, then their volumes are equal.

ANSWER: a. Sample answer:

b. The volumes are the same. The volume of a pyramid equals one third times the base area times ANSWER: the height. So, if the base areas of two pyramids are a. Sample answer: equal and their heights are equal, then their volumes are equal. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volumes will only be equal when the radius of The volume of a cone with radius r and height h the cone is equal to or when . equals the volume of a prism with height h. Therefore, the statement is true sometimes if the SOLUTION: base area of the cone is 3 times as great as the base The volume of a cone with a radius r and height h is area of the prism. For example, if the base of the prism has an area of 10 square units, then its volume . The volume of a prism with a height of is 10h cubic units. So, the cone must have a base h is where B is the area of the base of the area of 30 square units so that its volume is prism. Set the volumes equal. or 10h cubic units.

or 10h cubic units.

ANSWER: SOLUTION: Sometimes; the statement is true if the base area of the cone is 3 times as great as the base area of the The slant height is used for surface area, but the prism. For example, if the base of the prism has an height is used for volume. For this cone, the slant area of 10 square units, then its volume is 10h cubic height of 13 is provided, and we need to calculate the

units. So, the cone must have a base area of 30 height before we can calculate the volume.

square units so that its volume is or 10h Alexandra incorrectly used the slant height. cubic units. ANSWER: 37. ERROR ANALYSIS Alexandra and Cornelio are Cornelio; Alexandra incorrectly used the slant height. calculating the volume of the cone below. Is either of them correct? Explain your answer. 38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. SOLUTION: 3 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height. SOLUTION: The slant height is used for surface area, but the ANSWER: height is used for volume. For this cone, the slant 1704 cm3; The volume of a cylinder is three times as height of 13 is provided, and we need to calculate the much as the volume of a cone with the same radius height before we can calculate the volume. and height.

3 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

ANSWER: Set the base areas of the prism and pyramid, and 3 make the height of the pyramid equal to 3 times the 1704 cm ; The volume of a cylinder is three times as height of the prism. much as the volume of a cone with the same radius and height. Sample answer: A square pyramid with a base area of 16 and a OPEN ENDED 39. Give an example of a pyramid and height of 12, a prism with a square base area of 16 a prism that have the same base and the same and a height of 4. volume. Explain your reasoning. SOLUTION: ANSWER: The formula for volume of a prism is V = Bh and the Sample answer: A square pyramid with a base area formula for the volume of a pyramid is one-third of of 16 and a height of 12, a prism with a square base that. So, if a pyramid and prism have the same base, area of 16 and a height of 4; if a pyramid and prism then in order to have the same volume, the height of have the same base, then in order to have the same the pyramid must be 3 times as great as the height of volume, the height of the pyramid must be 3 times as

the prism. great as the height of the prism.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a Set the base areas of the prism and pyramid, and pyramid is one third the volume of a prism that has make the height of the pyramid equal to 3 times the the same height and base area. The volume of a height of the prism. cone is one third the volume of a cylinder that has the same height and base area. Sample answer: A square pyramid with a base area of 16 and a ANSWER: height of 12, a prism with a square base area of 16 To find the volume of each solid, you must know the and a height of 4. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has ANSWER: the same height and base area. The volume of a Sample answer: A square pyramid with a base area cone is one third the volume of a cylinder that has the of 16 and a height of 12, a prism with a square base same height and base area. area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same 41. A conical sand toy has the dimensions as shown volume, the height of the pyramid must be 3 times as below. How many cubic centimeters of sand will it great as the height of the prism. hold when it is filled to the top?

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders.

SOLUTION: A 12π To find the volume of each solid, you must know the B 15π area of the base and the height. The volume of a pyramid is one third the volume of a prism that has C the same height and base area. The volume of a

cone is one third the volume of a cylinder that has the D same height and base area.

ANSWER: SOLUTION: To find the volume of each solid, you must know the Use the Pythagorean Theorem to find the radius r of

area of the base and the height. The volume of a the cone. pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

A 12π B 15π

C So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where D r is the radius of the base and h is the height of the SOLUTION: cone. Use the Pythagorean Theorem to find the radius r of the cone.

Therefore, the correct choice is A.

ANSWER: A

Therefore, the correct choice is A.

12-5ANSWER: Volumes of Pyramids and Cones ANSWER: A 5.5 ft

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

SOLUTION: ANSWER: Possible outcomes: {6 red, 4 blue, 5 orange, 10 5.5 ft green} Number of possible outcomes : 25 43. PROBABILITY A spinner has sections colored Favorable outcomes: {5 orange} red, blue, orange, and green. The table below shows Number of favorable outcomes: 5 the results of several spins. What is the experimental probability of the spinner landing on orange?

F So, the correct choice is F. ANSWER: G F

H 44. SAT/ACT For all

J A –8 B x – 4 SOLUTION: C Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} D Number of possible outcomes : 25 Favorable outcomes: {5 orange} E Number of favorable outcomes: 5 SOLUTION:

eSolutions Manual - Powered by Cognero Page 19

So, the correct choice is E.

So, the correct choice is F. ANSWER: E ANSWER: F Find the volume of each prism.

44. SAT/ACT For all A –8 45. B x – 4 C SOLUTION: The volume of a prism is , where B is the D area of the base and h is the height of the prism. The E base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches. SOLUTION:

So, the correct choice is E.

3 ANSWER: Therefore, the volume of the prism is 1008 in . E ANSWER: Find the volume of each prism. 1008 in3

45.

SOLUTION: 46. The volume of a prism is , where B is the area of the base and h is the height of the prism. The SOLUTION: base of this prism is a rectangle with a length of 14 The volume of a prism is , where B is the inches and a width of 12 inches. The height h of the area of the base and h is the height of the prism. The prism is 6 inches. base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

3 Therefore, the volume of the prism is 1008 in .

ANSWER:

3 1008 in

So, the height of the triangle is 12 feet. Find the area 46. of the triangle. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle. So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 1140 ft So, the height of the triangle is 12 feet. Find the area of the triangle.

47. SOLUTION: The volume of a prism is , where B is the So, the area of the base B is 60 ft2. area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 The height h of the prism is 19 feet. meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

3 Therefore, the volume of the prism is 1140 ft . Therefore, the volume of the prism is about 426,437.6 ANSWER: m3. 3 1140 ft ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin 47. allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface SOLUTION: area of the bin with a conical top and bottom. Write The volume of a prism is , where B is the the exact answer and the answer rounded to the area of the base and h is the height of the prism. The nearest square foot. base of this prism is a rectangle with a length of 79.4 Refer to the photo on Page 863. meters and a width of 52.5 meters. The height of the prism is 102.3 meters. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. Therefore, the volume of the prism is about 426,437.6 m3. Find the slant height of the conical top.

ANSWER: 3 426,437.6 m

Find the slant height of the conical top.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the Find the slant height of the conical bottom. cylinder + Surface area of the conical top + surface The height of the conical bottom is 28 – (5 + 12 + 2) area of the conical bottom. or 9 ft.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant 49. height of the cylinder. SOLUTION:

Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) Surface area of the bin = Surface area of the = 50 cylinder + Surface area of the conical top + surface area of the conical bottom.

ANSWER: 2 ANSWER: 30.4 cm

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

ANSWER: 30.4 cm2

50.

SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Now find the area of the hexagon.

ANSWER:

2 54.4 in

51. SOLUTION: The shaded area is the area of the equilateral triangle Now find the area of the hexagon. less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

ANSWER: 2 54.4 in

The equilateral triangle can be divided into three 51. isosceles triangles with central angles of or SOLUTION: 120°, so the angle in the triangle created by the The shaded area is the area of the equilateral triangle height of the isosceles triangle is 60° and the base is less the area of the inscribed circle. half the length b of the side of the equilateral triangle.

Find the area of the circle with a radius of 3.6 feet.

Use a trigonometric ratio to find the value of b. So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

The equilateral triangle can be divided into three isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is So, the area of the equilateral triangle is about 67.3 half the length b of the side of the equilateral triangle. square feet.

Subtract to find the area of shaded region.

Use a trigonometric ratio to find the value of b. Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Subtract to find the area of shaded region. circle.

Therefore, the area of the shaded region is about 26.6 ft2. Divide the equilateral triangle into three isosceles ANSWER: triangles with central angles of or 120°. The 2 26.6 ft angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Use trigonometric ratios to find the values of b and Therefore, the area of the shaded region is about h. 168.2 mm2.

ANSWER: 2 168.2 mm

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

Find the volume of each pyramid.

SOLUTION: ANSWER: 3 The volume of a pyramid is , where B is the 132 cm area of the base and h is the height of the pyramid. 3. a rectangular pyramid with a height of 5.2 meters The base of this pyramid is a right triangle with legs and a base 8 meters by 4.5 meters of 9 inches and 5 inches and the height of the pyramid is 10 inches. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: 3 75 in

ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths 2. SOLUTION:

SOLUTION: The volume of a pyramid is , where B is the The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 area of the base and h is the height of the pyramid. meters. The height of the pyramid is 14 meters. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth.

ANSWER: 132 cm3

3. a rectangular pyramid with a height of 5.2 meters 5. and a base 8 meters by 4.5 meters SOLUTION: SOLUTION: The volume of a circular cone is , or The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. , where B is the area of the base, h is the The base of this pyramid is a rectangle with a length

of 8 meters and a width of 4.5 meters. The height of height of the cone, and r is the radius of the base. the pyramid is 5.2 meters. Since the diameter of this cone is 7 inches, the radius is or 3.5 inches. The height of the cone is 4 inches.

ANSWER: 3 62.4 m ANSWER: 3 51.3 in 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 6. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. SOLUTION:

Use trigonometry to find the radius r.

ANSWER: 298.7 m3 The volume of a circular cone is , or Find the volume of each cone. Round to the nearest tenth. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

5. SOLUTION:

The volume of a circular cone is , or ANSWER: , where B is the area of the base, h is the 168.1 cm3 height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius 7. an oblique cone with a height of 10.5 millimeters and is or 3.5 inches. The height of the cone is 4 inches. a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the ANSWER: height is 10.5 millimeters. 3 51.3 in

6. ANSWER: 3 SOLUTION: 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

Use trigonometry to find the radius r.

The volume of a circular cone is , or Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. , where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

So, the height of the cone is 20 meters.

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION: ANSWER: The volume of a circular cone is , or 4712.4 m3

, where B is the area of the base, h is the 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical height of the cone, and r is the radius of the base. building. If the height is 100 feet and the area of the The radius of this cone is 1.6 millimeters and the base is about 15,400 square feet, find the volume of height is 10.5 millimeters. air that the heating and cooling systems would have to accommodate. Round to the nearest tenth.

SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. ANSWER: For this cone, the area of the base is 15,400 square 3 feet and the height is 100 feet. 28.1 mm 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

ANSWER: 3 513,333.3 ft

Use the Pythagorean Theorem to find the height h of CCSS SENSE-MAKING Find the volume of the cone. Then find its volume. each pyramid. Round to the nearest tenth if necessary.

10. So, the height of the cone is 20 meters. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 4712.4 m3

9. MUSEUMS The sky dome of the National Corvette ANSWER: Museum in Bowling Green, Kentucky, is a conical 605 in3 building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where B 11. is the area of the base and h is the height of the SOLUTION: cone. For this cone, the area of the base is 15,400 square The volume of a pyramid is , where B is the feet and the height is 100 feet. area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 3 513,333.3 ft 3 105.8 mm CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

12. SOLUTION:

10. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid.

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

ANSWER: 482.1 m3 ANSWER: 605 in3

13. SOLUTION:

11. The volume of a pyramid is , where B is the SOLUTION: base and h is the height of the pyramid.

The volume of a pyramid is , where B is the The base is a hexagon, so we need to make a right tri area of the base and h is the height of the pyramid. determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl

90° triangle.

ANSWER: 3 105.8 mm

12. The apothem is .

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 233.8 cm ANSWER: 3 14. a pentagonal pyramid with a base area of 590 square 482.1 m feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13.

SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri ANSWER: determine the apothem. The interior angles of the he 1376.7 ft3 120°. The radius bisects the angle, so the right triangl 90° triangle. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The apothem is .

ANSWER: 3 233.8 cm The volume of a pyramid is , where B is the 14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet area of the base and h is the height of the pyramid.

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has ANSWER: a volume of 144 cubic centimeters. Find the height. 1376.7 ft3 SOLUTION: 15. a triangular pyramid with a height of 4.8 centimeters The base of the pyramid is a right triangle with a leg and a right triangle base with a leg 5 centimeters and of 8 centimeters and a hypotenuse of 10 centimeters. hypotenuse 10.2 centimeters Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the SOLUTION: triangle. Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

The volume of a pyramid is , where B is the 2 area of the base and h is the height of the pyramid. So, the area of the base B is 24 cm .

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has Therefore, the height of the triangular pyramid is 18 a volume of 144 cubic centimeters. Find the height. cm.

SOLUTION: ANSWER: The base of the pyramid is a right triangle with a leg 18 cm of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a Find the volume of each cone. Round to the of the right triangle and then find the area of the nearest tenth. triangle.

17. SOLUTION:

The volume of a circular cone is ,

where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches. The length of the other leg of the right triangle is 6 cm.

3 Therefore, the volume of the cone is about 235.6 in .

So, the area of the base B is 24 cm2. ANSWER: 235.6 in3 Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the Therefore, the height of the triangular pyramid is 18 cone. The radius of this cone is 4.2 centimeters and cm. the height is 7.3 centimeters.

ANSWER:

18 cm Find the volume of each cone. Round to the nearest tenth.

Therefore, the volume of the cone is about 134.8 3 cm . 17. SOLUTION: ANSWER: 134.8 cm3 The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 19. inches. SOLUTION:

3 Therefore, the volume of the cone is about 235.6 in . Use a trigonometric ratio to find the height h of the cone. ANSWER: 235.6 in3

The volume of a circular cone is , where r is the radius of the base and h is the height of the 18. cone. The radius of this cone is 8 centimeters.

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the volume of the cone is about 1473.1 3 cm .

ANSWER: 1473.1 cm3

Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 20. 134.8 cm3 SOLUTION:

19.

SOLUTION: Use trigonometric ratios to find the height h and the radius r of the cone.

Use a trigonometric ratio to find the height h of the cone. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an Therefore, the volume of the cone is about 1473.1 altitude of 16 inches 3 cm . SOLUTION: ANSWER: The volume of a circular cone is , where 1473.1 cm3 r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

20. SOLUTION:

Therefore, the volume of the cone is about 1072.3 3 in .

Use trigonometric ratios to find the height h and the ANSWER: 3 radius r of the cone. 1072.3 in

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean

The volume of a circular cone is , where Theorem to find the height h of the cone. r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where 3 r is the radius of the base and h is the height of the Therefore, the volume of the cone is about 5.8 cm . cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches. ANSWER: 3 5.8 cm

Therefore, the volume of the cone is about 1072.3 SOLUTION: 3 in . The volume of a circular cone is , where ANSWER: r is the radius of the base and h is the height of the 1072.3 in3 cone. Since the diameter of the cone is 8 centimeters, the radius is or 4 centimeters. The 22. a right cone with a slant height of 5.6 centimeters height of the cone is 14 centimeters. and a radius of 1 centimeter

SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 3 5.8 cm ANSWER: 3 23. SNACKS Approximately how many cubic 42,000,000 ft centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base 25. GARDENING The greenhouse is a regular diameter of 8 centimeters? Round to the nearest octagonal pyramid with a height of 5 feet. The base tenth. has side lengths of 2 feet. What is the volume of the greenhouse? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle 3 Therefore, the paper cone will hold about 234.6 cm below is 22.5°. of roasted peanuts. ANSWER: 234.6 cm3

24. CCSS MODELING The Pyramid Arena in Memphis, Tennessee, is the third largest pyramid in the world. It is approximately 350 feet tall, and its

square base is 600 feet wide. Find the volume of this Use a trigonometric ratio to find the apothem a. pyramid. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. The height of this pyramid is 5 feet.

ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular Therefore, the volume of the greenhouse is about octagonal pyramid with a height of 5 feet. The base 3 has side lengths of 2 feet. What is the volume of the 32.2 ft . greenhouse? ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

SOLUTION:

The volume of a pyramid is , where B is the 26. area of the base and h is the height of the pyramid. SOLUTION: The base of the pyramid is a regular octagon with Volume of the solid given = Volume of the small sides of 2 feet. A central angle of the octagon is cone + Volume of the large cone or 45°, so the angle formed in the triangle below is 22.5°.

Use a trigonometric ratio to find the apothem a.

ANSWER: 471.2 in3

The height of this pyramid is 5 feet.

27. SOLUTION:

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 ANSWER: 32.2 ft 3 3190.6 m Find the volume of each solid. Round to the nearest tenth.

28. 26. SOLUTION: SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) ANSWER: required to heat the building. For new construction 471.2 in3 with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

27. SOLUTION: SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

ANSWER: 3 3190.6 m The volume of the ceiling is

28. The total volume is therefore 5000 + 1666.67 = SOLUTION: 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of ANSWER: two congruent rectangular pyramids. The base of 7698.5 cm3 each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches. 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she Determine the volume of the model. Explain why needs to determine the BTUs (British Thermal Units) knowing the volume is helpful in this situation. required to heat the building. For new construction with good insulation, there should be 2 BTUs per SOLUTION: cubic foot. What size unit does Sam need to purchase? The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

SOLUTION: The building can be broken down into the rectangular It tells Marta how much clay is needed to make the base and the pyramid ceiling. The volume of the base model. is ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. The volume of the ceiling is Describe how each change affects the volume of the cone. a. The height is doubled. b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the The total volume is therefore 5000 + 1666.67 = values. 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on a. Double h. rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches.

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. The volume is doubled. SOLUTION: b. Double r. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

2 The volume is multiplied by 2 or 4.

It tells Marta how much clay is needed to make the c. Double r and h. model.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. 3 Describe how each change affects the volume of the volume is multiplied by 2 or 8. cone. a. The height is doubled. ANSWER: b. The radius is doubled. a. The volume is doubled. 2 c. Both the radius and the height are doubled. b. The volume is multiplied by 2 or 4. 3 SOLUTION: c. The volume is multiplied by 2 or 8. Find the volume of the original cone. Then alter the Find each measure. Round to the nearest tenth values. if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. a. Double h. Let s be the side length of the base.

The volume is doubled.

b. Double r.

The side length of the base is 15 cm.

ANSWER: 15 cm 2 The volume is multiplied by 2 or 4. 33. The volume of a cone is 196π cubic inches and the c. Double r and h. height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. volume is multiplied by 23 or 8. Since the diameter is 8 centimeters, the radius is 4 centimeters. ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the The diameter is 2(7) or 14 inches.

area of the base and h is the height of the pyramid. ANSWER:

Let s be the side length of the base. 14 in.

34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The side length of the base is 15 cm. The lateral area of a cone is , where r is ANSWER: the radius and is the slant height of the cone. 15 cm Replace L with 71.6 and with 6, then solve for the radius r. 33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the So, the radius is about 3.8 millimeters.

height of the cone, and r is the radius of the base. Use the Pythagorean Theorem to find the height of Since the diameter is 8 centimeters, the radius is 4 the cone. centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the

The diameter is 2(7) or 14 inches. cone. ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION: Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with The lateral area of a cone is , where r is different bases that have a height of 10 centimeters the radius and is the slant height of the cone. and a base area of 24 square centimeters. Replace L with 71.6 and with 6, then solve for the b. VERBAL What is true about the volumes of the radius r. two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24. So, the radius is about 3.8 millimeters. Sample answer: Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

are equal. Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 3 70.2 mm 35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters

and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the

two pyramids that you drew? Explain. c. If the base area is multiplied by 5, the volume is c. ANALYTICAL Explain how multiplying the base multiplied by 5. If the height is multiplied by 5, the area and/or the height of the pyramid by 5 affects the volume is multiplied by 5. If both the base area and volume of the pyramid. the height are multiplied by 5, the volume is multiplied SOLUTION: by 5 · 5 or 25. a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

ANSWER: a. Sample answer:

or 10h cubic units.

b. The volumes are the same. The volume of a pyramid equals one third times the base area times ANSWER: the height. So, if the base areas of two pyramids are Sometimes; the statement is true if the base area of equal and their heights are equal, then their volumes the cone is 3 times as great as the base area of the are equal. prism. For example, if the base of the prism has an c. If the base area is multiplied by 5, the volume is area of 10 square units, then its volume is 10h cubic multiplied by 5. If the height is multiplied by 5, the units. So, the cone must have a base area of 30 volume is multiplied by 5. If both the base area and square units so that its volume is or 10h the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. cubic units.

36. CCSS ARGUMENTS Determine whether the 37. ERROR ANALYSIS Alexandra and Cornelio are following statement is sometimes, always, or never calculating the volume of the cone below. Is either of true. Justify your reasoning. them correct? Explain your answer. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height.

The volumes will only be equal when the radius of 38. REASONING A cone has a volume of 568 cubic the cone is equal to or when . centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain Therefore, the statement is true sometimes if the your reasoning. base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the SOLUTION: prism has an area of 10 square units, then its volume 3 is 10h cubic units. So, the cone must have a base 1704 cm ; The formula for the volume of a cylinder area of 30 square units so that its volume is is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times or 10h cubic units. as much as the volume of a cone with the same radius and height. ANSWER: Sometimes; the statement is true if the base area of ANSWER: the cone is 3 times as great as the base area of the 1704 cm3; The volume of a cylinder is three times as prism. For example, if the base of the prism has an much as the volume of a cone with the same radius area of 10 square units, then its volume is 10h cubic and height. units. So, the cone must have a base area of 30 39. OPEN ENDED Give an example of a pyramid and square units so that its volume is or 10h a prism that have the same base and the same cubic units. volume. Explain your reasoning. SOLUTION: 37. ERROR ANALYSIS Alexandra and Cornelio are The formula for volume of a prism is V = Bh and the calculating the volume of the cone below. Is either of formula for the volume of a pyramid is one-third of them correct? Explain your answer. that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

SOLUTION: Set the base areas of the prism and pyramid, and The slant height is used for surface area, but the make the height of the pyramid equal to 3 times the height is used for volume. For this cone, the slant height of the prism. height of 13 is provided, and we need to calculate the height before we can calculate the volume. Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 Alexandra incorrectly used the slant height. and a height of 4.

ANSWER: ANSWER: Cornelio; Alexandra incorrectly used the slant height. Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base 38. REASONING A cone has a volume of 568 cubic area of 16 and a height of 4; if a pyramid and prism centimeters. What is the volume of a cylinder that have the same base, then in order to have the same has the same radius and height as the cone? Explain volume, the height of the pyramid must be 3 times as your reasoning. great as the height of the prism.

SOLUTION: 40. WRITING IN MATH Compare and contrast 3 1704 cm ; The formula for the volume of a cylinder finding volumes of pyramids and cones with finding is V= Bh, while the formula for the volume of a cone volumes of prisms and cylinders. is V = Bh. The volume of a cylinder is three times SOLUTION: as much as the volume of a cone with the same To find the volume of each solid, you must know the radius and height. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has ANSWER: the same height and base area. The volume of a 1704 cm3; The volume of a cylinder is three times as cone is one third the volume of a cylinder that has the much as the volume of a cone with the same radius same height and base area. and height. ANSWER: 39. OPEN ENDED Give an example of a pyramid and To find the volume of each solid, you must know the a prism that have the same base and the same area of the base and the height. The volume of a volume. Explain your reasoning. pyramid is one third the volume of a prism that has the same height and base area. The volume of a SOLUTION: cone is one third the volume of a cylinder that has the The formula for volume of a prism is V = Bh and the same height and base area. formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, 41. A conical sand toy has the dimensions as shown then in order to have the same volume, the height of below. How many cubic centimeters of sand will it the pyramid must be 3 times as great as the height of hold when it is filled to the top? the prism.

A 12π B 15π

C Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the D height of the prism.

Sample answer: SOLUTION: A square pyramid with a base area of 16 and a Use the Pythagorean Theorem to find the radius r of height of 12, a prism with a square base area of 16 the cone. and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a So, the radius of the cone is 3 centimeters. pyramid is one third the volume of a prism that has the same height and base area. The volume of a The volume of a circular cone is , where cone is one third the volume of a cylinder that has the r is the radius of the base and h is the height of the same height and base area. cone.

ANSWER:

To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area. Therefore, the correct choice is A. 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it ANSWER: hold when it is filled to the top? A

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air,

how tall is the tent’s center pole? A 12π B 15π SOLUTION:

C The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. D

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where

r is the radius of the base and h is the height of the F cone.

Therefore, the correct choice is A. SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 ANSWER: green} A Number of possible outcomes : 25 Favorable outcomes: {5 orange} 42. SHORT RESPONSE Brooke is buying a tent that Number of favorable outcomes: 5 is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. So, the correct choice is F.

ANSWER: F

44. SAT/ACT For all

A –8 B x – 4 C ANSWER: D 5.5 ft E 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows SOLUTION: the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the correct choice is E.

ANSWER: F E

G Find the volume of each prism.

J 45. SOLUTION: SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 The volume of a prism is , where B is the green} area of the base and h is the height of the prism. The Number of possible outcomes : 25 base of this prism is a rectangle with a length of 14 Favorable outcomes: {5 orange} inches and a width of 12 inches. The height h of the

Number of favorable outcomes: 5 prism is 6 inches.

3 Therefore, the volume of the prism is 1008 in . So, the correct choice is F. ANSWER: ANSWER: 12-5 Volumes of Pyramids and Cones 1008 in3 F

44. SAT/ACT For all

A –8 B x – 4 46. C SOLUTION: D The volume of a prism is , where B is the E area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will SOLUTION: bisect the base. Use the Pythagorean Theorem to find the height of the triangle.

So, the correct choice is E.

ANSWER: E

Find the volume of each prism.

45. So, the height of the triangle is 12 feet. Find the area SOLUTION: of the triangle. The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1008 in .

ANSWER:

1008 in3 Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 1140 ft

46. eSolutions Manual - Powered by Cognero Page 20 SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The 47. base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will SOLUTION: bisect the base. Use the Pythagorean Theorem to The volume of a prism is , where B is the find the height of the triangle. area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER:

3 So, the height of the triangle is 12 feet. Find the area 426,437.6 m of the triangle. 48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. 2 So, the area of the base B is 60 ft . Refer to the photo on Page 863.

The height h of the prism is 19 feet. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. Therefore, the volume of the prism is 1140 ft3. Find the slant height of the conical top. ANSWER: 3 1140 ft

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination The formula for finding the surface area of a cylinder hopper cone and bin used by farmers to store grain is , where is r is the radius and h is the slant after harvest. The cone at the bottom of the bin height of the cylinder. allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the Surface area of the bin = Surface area of the nearest square foot. cylinder + Surface area of the conical top + surface Refer to the photo on Page 863. area of the conical bottom. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone. ANSWER:

Find the slant height of the conical top. Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

ANSWER: 30.4 cm2

50. SOLUTION: The formula for finding the surface area of a cylinder Area of the shaded region = Area of the circle – is , where is r is the radius and h is the slant Area of the hexagon height of the cylinder.

Surface area of the bin = Surface area of the

cylinder + Surface area of the conical top + surface A regular hexagon has 6 sides, so the measure of the area of the conical bottom. interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle –

Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: Now find the area of the hexagon. 2 30.4 cm

50.

SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

ANSWER: 2 54.4 in

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the 51. length of the apothem. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

ANSWER: Now use the area of the isosceles triangles to find 2 54.4 in the area of the equilateral triangle.

51. SOLUTION: So, the area of the equilateral triangle is about 67.3 The shaded area is the area of the equilateral triangle square feet. less the area of the inscribed circle.

Subtract to find the area of shaded region. Find the area of the circle with a radius of 3.6 feet.

Therefore, the area of the shaded region is about So, the area of the circle is about 40.7 square feet. 2 26.6 ft .

Next, find the area of the equilateral triangle. ANSWER: 2 26.6 ft

52.

The equilateral triangle can be divided into three SOLUTION: The area of the shaded region is the area of the isosceles triangles with central angles of or circumscribed circle minus the area of the equilateral 120°, so the angle in the triangle created by the triangle plus the area of the inscribed circle. Let b height of the isosceles triangle is 60° and the base is represent the length of each side of the equilateral half the length b of the side of the equilateral triangle. triangle and h represent the radius of the inscribed circle.

Use a trigonometric ratio to find the value of b.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown Now use the area of the isosceles triangles to find below. the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 Use trigonometric ratios to find the values of b and

square feet. h.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 2 26.6 ft . The area of the equilateral triangle will equal three ANSWER: times the area of any of the isosceles triangles. Find 2 26.6 ft the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b Therefore, the area of the shaded region is about represent the length of each side of the equilateral 168.2 mm2. triangle and h represent the radius of the inscribed circle. ANSWER: 2 168.2 mm

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

Find the volume of each pyramid.

ANSWER: 132 cm3

ANSWER: 3 ANSWER: 62.4 m 3 75 in 4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

3 298.7 m Find the volume of each cone. Round to the nearest tenth.

ANSWER: 3 51.3 in

ANSWER: 3 62.4 m

meters. The height of the pyramid is 14 meters. Use trigonometry to find the radius r.

The volume of a circular cone is , or

5. ANSWER: 3 SOLUTION: 168.1 cm

The volume of a circular cone is , or 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters , where B is the area of the base, h is the SOLUTION:

height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

ANSWER: 3 51.3 in

ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius 6. of 15 meters SOLUTION: SOLUTION:

Use trigonometry to find the radius r. Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters. So, the height of the cone is 20 meters.

is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters ANSWER: 3 SOLUTION: 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

10. SOLUTION:

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. So, the height of the cone is 20 meters.

ANSWER: 3 ANSWER: 605 in 4712.4 m3

ANSWER: 3 105.8 mm

ANSWER: 3 513,333.3 ft

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 482.1 m

ANSWER: 13. 605 in3 SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri 11. determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl SOLUTION: 90° triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 105.8 mm The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

ANSWER: The volume of a pyramid is , where B is the 482.1 m3 area of the base and h is the height of the pyramid.

13. SOLUTION: ANSWER: The volume of a pyramid is , where B is the 1376.7 ft3

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the ANSWER: 3 area of the base and h is the height of the pyramid. 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: The base of the pyramid is a right triangle with a leg of 8 centimeters and a hypotenuse of 10 centimeters. ANSWER: Use the Pythagorean Theorem to find the other leg a 1376.7 ft3 of the right triangle and then find the area of the triangle. 15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm. 3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h. 18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18 cm.

ANSWER: 18 cm Therefore, the volume of the cone is about 134.8 Find the volume of each cone. Round to the 3 nearest tenth. cm . ANSWER: 134.8 cm3

17. SOLUTION:

The volume of a circular cone is , 19. where r is the radius of the base and h is the height of the cone. SOLUTION:

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

Use a trigonometric ratio to find the height h of the cone.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: The volume of a circular cone is , where 235.6 in3 r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

18. SOLUTION:

1473.1 cm3

20. Therefore, the volume of the cone is about 134.8 SOLUTION: 3 cm .

ANSWER: 134.8 cm3

Use trigonometric ratios to find the height h and the radius r of the cone.

19. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Use a trigonometric ratio to find the height h of the cone.

2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

ANSWER: the radius is or 8 inches.

1473.1 cm3

20. SOLUTION: Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an 3 altitude of 16 inches Therefore, the volume of the cone is about 5.8 cm .

SOLUTION: ANSWER: The volume of a circular cone is , where 5.8 cm3

1072.3 in3

22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION:

SOLUTION: The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 3 Therefore, the volume of the cone is about 5.8 cm . 42,000,000 ft

ANSWER: 25. GARDENING The greenhouse is a regular 5.8 cm3 octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the 23. SNACKS Approximately how many cubic greenhouse? centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 3 234.6 cm Use a trigonometric ratio to find the apothem a.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

26.

SOLUTION: Volume of the solid given = Volume of the small SOLUTION: cone + Volume of the large cone

The volume of a pyramid is , where B is the

ANSWER: 471.2 in3

Use a trigonometric ratio to find the apothem a.

27. The height of this pyramid is 5 feet. SOLUTION:

ANSWER: Therefore, the volume of the greenhouse is about 3 3190.6 m 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth. 28. SOLUTION:

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

ANSWER: 471.2 in3

SOLUTION: The building can be broken down into the rectangular 27. base and the pyramid ceiling. The volume of the base is SOLUTION:

The volume of the ceiling is

ANSWER: 3 3190.6 m

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs. 28.

SOLUTION: ANSWER: 13,333 BTUs

It tells Marta how much clay is needed to make the model.

c. Both the radius and the height are doubled. The volume of the ceiling is SOLUTION: Find the volume of the original cone. Then alter the values.

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid. 2 The volume is multiplied by 2 or 4.

c. Double r and h.

It tells Marta how much clay is needed to make the model.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

a. Double h.

The volume is doubled.

b. Double r. The side length of the base is 15 cm. ANSWER: 15 cm

volume is multiplied by 23 or 8.

ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r. The side length of the base is 15 cm.

ANSWER: 15 cm

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base The lateral area of a cone is , where r is area and/or the height of the pyramid by 5 affects the the radius and is the slant height of the cone. volume of the pyramid. Replace L with 71.6 and with 6, then solve for the radius r. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

Sample answer:

ANSWER: a. Sample answer:

or 10h cubic units.

ANSWER: Sometimes; the statement is true if the base area of the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an b. The volumes are the same. The volume of a area of 10 square units, then its volume is 10h cubic pyramid equals one third times the base area times units. So, the cone must have a base area of 30 the height. So, if the base areas of two pyramids are square units so that its volume is or 10h equal and their heights are equal, then their volumes are equal. cubic units. c. If the base area is multiplied by 5, the volume is multiplied by 5. If the height is multiplied by 5, the 37. ERROR ANALYSIS Alexandra and Cornelio are volume is multiplied by 5. If both the base area and calculating the volume of the cone below. Is either of the height are multiplied by 5, the volume is multiplied them correct? Explain your answer. by 5 · 5 or 25.

36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of SOLUTION: h is where B is the area of the base of the The slant height is used for surface area, but the prism. Set the volumes equal. height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning.

them correct? Explain your answer.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

SOLUTION: Set the base areas of the prism and pyramid, and Use the Pythagorean Theorem to find the radius r of make the height of the pyramid equal to 3 times the the cone. height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding So, the radius of the cone is 3 centimeters.

The volume of a pyramid is , where B is the

A 12π area of the base and h is the height of the pyramid.

B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

F So, the radius of the cone is 3 centimeters.

G The volume of a circular cone is , where

r is the radius of the base and h is the height of the H cone.

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Therefore, the correct choice is A. Number of favorable outcomes: 5

ANSWER: A

42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? So, the correct choice is F. SOLUTION: ANSWER: The volume of a pyramid is , where B is the F area of the base and h is the height of the pyramid.

44. SAT/ACT For all

A –8 B x – 4 C D E

SOLUTION: ANSWER: 5.5 ft

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 1008 in3

So, the correct choice is F.

SOLUTION:

So, the correct choice is E.

ANSWER: E

Find the volume of each prism. So, the height of the triangle is 12 feet. Find the area of the triangle.

Therefore, the volume of the prism is 1140 ft3. 3 Therefore, the volume of the prism is 1008 in . ANSWER: ANSWER: 3 1140 ft 1008 in3

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin So, the height of the triangle is 12 feet. Find the area allows the grain to be emptied more easily. Use the of the triangle. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find So, the area of the base B is 60 ft2. the surface area of the cylinder and add them. The formula for finding the surface area of a cone is The height h of the prism is 19 feet. , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 12-5 Volumes of Pyramids and Cones 3 1140 ft

Find the slant height of the conical bottom. 47. The height of the conical bottom is 28 – (5 + 12 + 2) SOLUTION: or 9 ft.

48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain Surface area of the bin = Surface area of the after harvest. The cone at the bottom of the bin cylinder + Surface area of the conical top + surface allows the grain to be emptied more easily. Use the area of the conical bottom. dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the ANSWER: surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is Find the area of each shaded region. Polygons , where is r is the radius and l is the slant height in 50 - 52 are regular. of the cone.

Find the slant height of the conical top. 49.

SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

eSolutions Manual - Powered by Cognero Page 21

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) ANSWER: or 9 ft. 30.4 cm2

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Now find the area of the hexagon.

ANSWER: 30.4 cm2

50. SOLUTION: ANSWER: 2 Area of the shaded region = Area of the circle – 54.4 in Area of the hexagon

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

The equilateral triangle can be divided into three

Now find the area of the hexagon. Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

ANSWER: 2 54.4 in

So, the area of the equilateral triangle is about 67.3 square feet.

51. Subtract to find the area of shaded region. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet. Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral The equilateral triangle can be divided into three triangle and h represent the radius of the inscribed circle. isosceles triangles with central angles of or 120°, so the angle in the triangle created by the height of the isosceles triangle is 60° and the base is half the length b of the side of the equilateral triangle.

Divide the equilateral triangle into three isosceles triangles with central angles of or 120°. The

angle in the triangle formed by the height.h of the Use a trigonometric ratio to find the value of b. isosceles triangle is 60° and the base is half the

length of the side of the equilateral triangle as shown below.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and h.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region. Therefore, the area of the shaded region is about

2 26.6 ft . A(shaded region) = A(circumscribed circle) - A

ANSWER: (equilateral triangle) + A(inscribed circle) 2 26.6 ft

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

Find the volume of each pyramid.

1. ANSWER: 3 SOLUTION: 132 cm

ANSWER: 3 75 in ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth.

ANSWER: 3 132 cm 5. 3. a rectangular pyramid with a height of 5.2 meters SOLUTION: and a base 8 meters by 4.5 meters The volume of a circular cone is , or SOLUTION:

ANSWER: ANSWER: 3 3 51.3 in 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

Use trigonometry to find the radius r.

5. SOLUTION:

The volume of a circular cone is , or ANSWER: 168.1 cm3 , where B is the area of the base, h is the 7. an oblique cone with a height of 10.5 millimeters and height of the cone, and r is the radius of the base. a radius of 1.6 millimeters Since the diameter of this cone is 7 inches, the radius SOLUTION: is or 3.5 inches. The height of the cone is 4 inches. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

ANSWER: 3 51.3 in

ANSWER: 3 6. 28.1 mm

SOLUTION: 8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

Use trigonometry to find the radius r.

Use the Pythagorean Theorem to find the height h of The volume of a circular cone is , or the cone. Then find its volume.

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

So, the height of the cone is 20 meters.

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters ANSWER: SOLUTION: 4712.4 m3

The volume of a circular cone is , or 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical , where B is the area of the base, h is the building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of height of the cone, and r is the radius of the base. air that the heating and cooling systems would have The radius of this cone is 1.6 millimeters and the to accommodate. Round to the nearest tenth. height is 10.5 millimeters. SOLUTION:

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if Use the Pythagorean Theorem to find the height h of necessary. the cone. Then find its volume.

10. SOLUTION:

So, the height of the cone is 20 meters. The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

SOLUTION: 11. The volume of a circular cone is , where B SOLUTION:

is the area of the base and h is the height of the The volume of a pyramid is , where B is the cone. For this cone, the area of the base is 15,400 square area of the base and h is the height of the pyramid. feet and the height is 100 feet.

ANSWER: 3 ANSWER: 105.8 mm 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. 12. SOLUTION:

The volume of a pyramid is , where B is the 10. area of the base and h is the height of the pyramid.

SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 482.1 m3

ANSWER: 605 in3

13. SOLUTION:

The volume of a pyramid is , where B is the

11. base and h is the height of the pyramid.

SOLUTION: The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he The volume of a pyramid is , where B is the 120°. The radius bisects the angle, so the right triangl area of the base and h is the height of the pyramid. 90° triangle.

ANSWER: 3 105.8 mm

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square ANSWER: feet and an altitude of 7 feet 3 482.1 m SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid. ANSWER:

1376.7 ft3 The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 15. a triangular pyramid with a height of 4.8 centimeters 120°. The radius bisects the angle, so the right triangl and a right triangle base with a leg 5 centimeters and 90° triangle. hypotenuse 10.2 centimeters

SOLUTION: Find the height of the right triangle.

The apothem is .

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 3 235.6 in So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters. Therefore, the height of the triangular pyramid is 18 cm. ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth. Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 17. 134.8 cm3 SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. 19. Since the diameter of this cone is 10 inches, the SOLUTION: radius is or 5 inches. The height of the cone is 9 inches.

Use a trigonometric ratio to find the height h of the

cone. 3 Therefore, the volume of the cone is about 235.6 in . ANSWER: 235.6 in3

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

18. SOLUTION:

ANSWER: 1473.1 cm3

Therefore, the volume of the cone is about 134.8 3 cm . 20. ANSWER: SOLUTION: 134.8 cm3

19. Use trigonometric ratios to find the height h and the radius r of the cone. SOLUTION:

The volume of a circular cone is , where

Use a trigonometric ratio to find the height h of the r is the radius of the base and h is the height of the cone. cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. 3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an

altitude of 16 inches Therefore, the volume of the cone is about 1473.1 SOLUTION: 3 cm . The volume of a circular cone is , where ANSWER: r is the radius of the base and h is the height of the 1473.1 cm3 cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

20. SOLUTION:

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3 Use trigonometric ratios to find the height h and the radius r of the cone. 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER:

2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches

SOLUTION: 3 The volume of a circular cone is , where Therefore, the volume of the cone is about 5.8 cm . r is the radius of the base and h is the height of the ANSWER: cone. Since the diameter of this cone is 16 inches, 3 5.8 cm the radius is or 8 inches. 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

Therefore, the volume of the cone is about 1072.3 The volume of a circular cone is , where 3 in . r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 ANSWER: 3 centimeters, the radius is or 4 centimeters. The 1072.3 in height of the cone is 14 centimeters. 22. a right cone with a slant height of 5.6 centimeters and a radius of 1 centimeter SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

The volume of a pyramid is , where B is the

area of the base and h is the height of the pyramid.

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: ANSWER: 3 5.8 cm3 42,000,000 ft

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8

centimeters, the radius is or 4 centimeters. The SOLUTION: height of the cone is 14 centimeters. The volume of a pyramid is , where B is the

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

24. CCSS MODELING The Pyramid Arena in

Memphis, Tennessee, is the third largest pyramid in Use a trigonometric ratio to find the apothem a. the world. It is approximately 350 feet tall, and its square base is 600 feet wide. Find the volume of this pyramid. SOLUTION:

The volume of a pyramid is , where B is the The height of this pyramid is 5 feet. area of the base and h is the height of the pyramid.

ANSWER: 3 42,000,000 ft Therefore, the volume of the greenhouse is about 3 25. GARDENING The greenhouse is a regular 32.2 ft . octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the ANSWER: 3 greenhouse? 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

SOLUTION: 26. SOLUTION: The volume of a pyramid is , where B is the Volume of the solid given = Volume of the small area of the base and h is the height of the pyramid. cone + Volume of the large cone The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

Use a trigonometric ratio to find the apothem a. ANSWER:

471.2 in3

The height of this pyramid is 5 feet.

27. SOLUTION:

Therefore, the volume of the greenhouse is about 32.2 ft3. ANSWER: ANSWER: 3 3190.6 m 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

28. SOLUTION: 26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 7698.5 cm3

27. SOLUTION: SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

The volume of the ceiling is ANSWER: 3 3190.6 m

The total volume is therefore 5000 + 1666.67 = 3 28. 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is SOLUTION: 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay

model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total ANSWER: height will be 4 inches. 3 7698.5 cm Determine the volume of the model. Explain why 29. HEATING Sam is building an art studio in her knowing the volume is helpful in this situation. backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) SOLUTION: required to heat the building. For new construction with good insulation, there should be 2 BTUs per The volume of a pyramid is , where B is the cubic foot. What size unit does Sam need to area of the base and h is the height of the pyramid. purchase?

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER:

Determine the volume of the model. Explain why The volume is doubled. knowing the volume is helpful in this situation. b. Double r. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

2 The volume is multiplied by 2 or 4.

c. Double r and h. It tells Marta how much clay is needed to make the model.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

31. CHANGING DIMENSIONS A cone has a radius volume is multiplied by 23 or 8. of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the ANSWER: cone. a. The volume is doubled. 2 a. The height is doubled. b. The volume is multiplied by 2 or 4. b. The radius is doubled. 3 c. The volume is multiplied by 2 or 8. c. Both the radius and the height are doubled. SOLUTION: Find each measure. Round to the nearest tenth if necessary. Find the volume of the original cone. Then alter the 32. A square pyramid has a volume of 862.5 cubic

values. centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

a. Double h.

The volume is doubled.

b. Double r.

The side length of the base is 15 cm.

ANSWER: 15 cm

2 33. The volume of a cone is 196π cubic inches and the The volume is multiplied by 2 or 4. height is 12 inches. What is the diameter?

c. Double r and h. SOLUTION: The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters. 3 volume is multiplied by 2 or 8.

ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base.

SOLUTION: The diameter is 2(7) or 14 inches.

The volume of a pyramid is , where B is the ANSWER: 14 in. area of the base and h is the height of the pyramid. Let s be the side length of the base. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The lateral area of a cone is , where r is The side length of the base is 15 cm. the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the ANSWER: radius r. 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or So, the radius is about 3.8 millimeters.

, where B is the area of the base, h is the Use the Pythagorean Theorem to find the height of the cone. height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the

cone.

The diameter is 2(7) or 14 inches.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the

volume of the cone? Therefore, the volume of the cone is about 70.2 3 SOLUTION: mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. The lateral area of a cone is , where r is b. VERBAL What is true about the volumes of the the radius and is the slant height of the cone. two pyramids that you drew? Explain. Replace L with 71.6 and with 6, then solve for the c. ANALYTICAL Explain how multiplying the base radius r. area and/or the height of the pyramid by 5 affects the volume of the pyramid.

SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer: So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the

cone. b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with

different bases that have a height of 10 centimeters and a base area of 24 square centimeters. c. If the base area is multiplied by 5, the volume is b. VERBAL What is true about the volumes of the multiplied by 5. If the height is multiplied by 5, the two pyramids that you drew? Explain. volume is multiplied by 5. If both the base area and c. ANALYTICAL Explain how multiplying the base the height are multiplied by 5, the volume is multiplied area and/or the height of the pyramid by 5 affects the by 5 · 5 or 25. volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

ANSWER: a. Sample answer:

or 10h cubic units.

ANSWER: b. The volumes are the same. The volume of a Sometimes; the statement is true if the base area of pyramid equals one third times the base area times the cone is 3 times as great as the base area of the the height. So, if the base areas of two pyramids are prism. For example, if the base of the prism has an equal and their heights are equal, then their volumes area of 10 square units, then its volume is 10h cubic are equal. units. So, the cone must have a base area of 30 c. If the base area is multiplied by 5, the volume is square units so that its volume is or 10h multiplied by 5. If the height is multiplied by 5, the volume is multiplied by 5. If both the base area and cubic units. the height are multiplied by 5, the volume is multiplied by 5 · 5 or 25. 37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of 36. CCSS ARGUMENTS Determine whether the them correct? Explain your answer. following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h. SOLUTION: The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal. SOLUTION: The slant height is used for surface area, but the height is used for volume. For this cone, the slant height of 13 is provided, and we need to calculate the height before we can calculate the volume.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic

centimeters. What is the volume of a cylinder that The volumes will only be equal when the radius of has the same radius and height as the cone? Explain the cone is equal to or when . your reasoning. Therefore, the statement is true sometimes if the SOLUTION: base area of the cone is 3 times as great as the base 3 area of the prism. For example, if the base of the 1704 cm ; The formula for the volume of a cylinder prism has an area of 10 square units, then its volume is V= Bh, while the formula for the volume of a cone is 10h cubic units. So, the cone must have a base is V = Bh. The volume of a cylinder is three times area of 30 square units so that its volume is as much as the volume of a cone with the same radius and height. or 10h cubic units. ANSWER: ANSWER: 1704 cm3; The volume of a cylinder is three times as Sometimes; the statement is true if the base area of much as the volume of a cone with the same radius the cone is 3 times as great as the base area of the and height. prism. For example, if the base of the prism has an area of 10 square units, then its volume is 10h cubic 39. OPEN ENDED Give an example of a pyramid and units. So, the cone must have a base area of 30 a prism that have the same base and the same volume. Explain your reasoning. square units so that its volume is or 10h SOLUTION: cubic units. The formula for volume of a prism is V = Bh and the 37. ERROR ANALYSIS Alexandra and Cornelio are formula for the volume of a pyramid is one-third of calculating the volume of the cone below. Is either of that. So, if a pyramid and prism have the same base, them correct? Explain your answer. then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and

make the height of the pyramid equal to 3 times the SOLUTION: height of the prism. The slant height is used for surface area, but the height is used for volume. For this cone, the slant Sample answer: height of 13 is provided, and we need to calculate the A square pyramid with a base area of 16 and a height before we can calculate the volume. height of 12, a prism with a square base area of 16 and a height of 4.

Alexandra incorrectly used the slant height. ANSWER: Sample answer: A square pyramid with a base area ANSWER: of 16 and a height of 12, a prism with a square base Cornelio; Alexandra incorrectly used the slant height. area of 16 and a height of 4; if a pyramid and prism have the same base, then in order to have the same 38. REASONING A cone has a volume of 568 cubic volume, the height of the pyramid must be 3 times as centimeters. What is the volume of a cylinder that great as the height of the prism. has the same radius and height as the cone? Explain your reasoning. 40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding SOLUTION: volumes of prisms and cylinders. 3 1704 cm ; The formula for the volume of a cylinder SOLUTION: is V= Bh, while the formula for the volume of a cone To find the volume of each solid, you must know the is V = Bh. The volume of a cylinder is three times area of the base and the height. The volume of a as much as the volume of a cone with the same pyramid is one third the volume of a prism that has radius and height. the same height and base area. The volume of a cone is one third the volume of a cylinder that has the ANSWER: same height and base area. 1704 cm3; The volume of a cylinder is three times as much as the volume of a cone with the same radius ANSWER: and height. To find the volume of each solid, you must know the area of the base and the height. The volume of a 39. OPEN ENDED Give an example of a pyramid and pyramid is one third the volume of a prism that has a prism that have the same base and the same the same height and base area. The volume of a volume. Explain your reasoning. cone is one third the volume of a cylinder that has the same height and base area. SOLUTION: The formula for volume of a prism is V = Bh and the 41. A conical sand toy has the dimensions as shown formula for the volume of a pyramid is one-third of below. How many cubic centimeters of sand will it that. So, if a pyramid and prism have the same base, hold when it is filled to the top? then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

A 12π B 15π

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders.

same height and base area. ANSWER: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the Therefore, the correct choice is A. same height and base area. ANSWER: 41. A conical sand toy has the dimensions as shown A below. How many cubic centimeters of sand will it hold when it is filled to the top? 42. SHORT RESPONSE Brooke is buying a tent that is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION: A 12π B 15π The volume of a pyramid is , where B is the

C area of the base and h is the height of the pyramid.

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the radius of the cone is 3 centimeters.

The volume of a circular cone is , where F r is the radius of the base and h is the height of the cone. G

The volume of a pyramid is , where B is the So, the correct choice is F. area of the base and h is the height of the pyramid. ANSWER: F

44. SAT/ACT For all

So, the correct choice is E.

ANSWER: E

F Find the volume of each prism.

H 45.

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 3 So, the correct choice is F. 1008 in ANSWER: F

44. SAT/ACT For all

A –8 46. B x – 4 SOLUTION: C The volume of a prism is , where B is the area of the base and h is the height of the prism. The D base of this prism is an isosceles triangle with a base E of 10 feet and two legs of 13 feet. The height h will bisect the base. Use the Pythagorean Theorem to find the height of the triangle. SOLUTION:

So, the correct choice is E.

ANSWER: E Find the volume of each prism.

So, the height of the triangle is 12 feet. Find the area 45. of the triangle.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1008 in .

ANSWER: Therefore, the volume of the prism is 1140 ft3. 1008 in3 ANSWER: 3 1140 ft

46. SOLUTION: 47. The volume of a prism is , where B is the area of the base and h is the height of the prism. The SOLUTION: base of this prism is an isosceles triangle with a base The volume of a prism is , where B is the of 10 feet and two legs of 13 feet. The height h will area of the base and h is the height of the prism. The bisect the base. Use the Pythagorean Theorem to base of this prism is a rectangle with a length of 79.4

find the height of the triangle. meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m

So, the area of the base B is 60 ft2. SOLUTION: To find the entire surface area of the bin, find the The height h of the prism is 19 feet. surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top. Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 1140 ft

47. SOLUTION: Find the slant height of the conical bottom. The volume of a prism is , where B is the The height of the conical bottom is 28 – (5 + 12 + 2) area of the base and h is the height of the prism. The or 9 ft. base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find the surface area of the cylinder and add them. The formula for finding the surface area of a cone is ANSWER: , where is r is the radius and l is the slant height of the cone.

Find the area of each shaded region. Polygons

Find the slant height of the conical top. in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

ANSWER: 30.4 cm2

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

A regular hexagon has 6 sides, so the measure of the Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface interior angle is . The apothem bisects the area of the conical bottom. angle, so we use 30° when using trig to find the length of the apothem.

ANSWER: 12-5 Volumes of Pyramids and Cones

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION:

Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Now find the area of the hexagon. ANSWER:

30.4 cm2

50.

SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon ANSWER: 2 54.4 in

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle. eSolutions Manual - Powered by Cognero Page 22

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find ANSWER: the area of the equilateral triangle. 2 54.4 in

Find the area of the circle with a radius of 3.6 feet.

Therefore, the area of the shaded region is about 2 26.6 ft . So, the area of the circle is about 40.7 square feet. ANSWER: 2 Next, find the area of the equilateral triangle. 26.6 ft

52. SOLUTION: The area of the shaded region is the area of the The equilateral triangle can be divided into three circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b isosceles triangles with central angles of or represent the length of each side of the equilateral 120°, so the angle in the triangle created by the triangle and h represent the radius of the inscribed height of the isosceles triangle is 60° and the base is circle. half the length b of the side of the equilateral triangle.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and h. So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 2 The area of the equilateral triangle will equal three 26.6 ft . times the area of any of the isosceles triangles. Find the area of the shaded region. ANSWER:

2 26.6 ft A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

52.

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

Find the volume of each pyramid.

ANSWER: 1. 132 cm3 SOLUTION: 3. a rectangular pyramid with a height of 5.2 meters The volume of a pyramid is , where B is the and a base 8 meters by 4.5 meters area of the base and h is the height of the pyramid. SOLUTION: The base of this pyramid is a right triangle with legs of 9 inches and 5 inches and the height of the The volume of a pyramid is , where B is the pyramid is 10 inches. area of the base and h is the height of the pyramid. The base of this pyramid is a rectangle with a length of 8 meters and a width of 4.5 meters. The height of the pyramid is 5.2 meters.

ANSWER: ANSWER: 3 75 in 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the 2. area of the base and h is the height of the pyramid. SOLUTION: The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters. The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a regular pentagon with sides of 4.4 centimeters and an apothem of 3 centimeters. The height of the pyramid is 12 centimeters.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth.

ANSWER: 5. 132 cm3 SOLUTION: 3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters The volume of a circular cone is , or SOLUTION: , where B is the area of the base, h is the The volume of a pyramid is , where B is the height of the cone, and r is the radius of the base. area of the base and h is the height of the pyramid. Since the diameter of this cone is 7 inches, the radius The base of this pyramid is a rectangle with a length is or 3.5 inches. The height of the cone is 4 inches. of 8 meters and a width of 4.5 meters. The height of

the pyramid is 5.2 meters.

ANSWER: 3 ANSWER: 51.3 in 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION: 6. The volume of a pyramid is , where B is the SOLUTION: area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

Use trigonometry to find the radius r.

The volume of a circular cone is , or ANSWER: 3 298.7 m , where B is the area of the base, h is the Find the volume of each cone. Round to the height of the cone, and r is the radius of the base. nearest tenth. The height of the cone is 11.5 centimeters.

5. SOLUTION: ANSWER: The volume of a circular cone is , or 168.1 cm3

, where B is the area of the base, h is the 7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters height of the cone, and r is the radius of the base. Since the diameter of this cone is 7 inches, the radius SOLUTION: is or 3.5 inches. The height of the cone is 4 inches. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

ANSWER: 3 51.3 in

ANSWER: 3 28.1 mm 6. 8. a cone with a slant height of 25 meters and a radius SOLUTION: of 15 meters SOLUTION:

Use trigonometry to find the radius r.

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

So, the height of the cone is 20 meters.

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters ANSWER: 4712.4 m3 SOLUTION: MUSEUMS The volume of a circular cone is , or 9. The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the , where B is the area of the base, h is the base is about 15,400 square feet, find the volume of height of the cone, and r is the radius of the base. air that the heating and cooling systems would have The radius of this cone is 1.6 millimeters and the to accommodate. Round to the nearest tenth. height is 10.5 millimeters. SOLUTION:

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION: ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

10. SOLUTION:

The volume of a pyramid is , where B is the So, the height of the cone is 20 meters. area of the base and h is the height of the pyramid.

ANSWER: ANSWER: 4712.4 m3 605 in3 9. MUSEUMS The sky dome of the National Corvette Museum in Bowling Green, Kentucky, is a conical building. If the height is 100 feet and the area of the base is about 15,400 square feet, find the volume of air that the heating and cooling systems would have to accommodate. Round to the nearest tenth. 11. SOLUTION: SOLUTION: The volume of a circular cone is , where B The volume of a pyramid is , where B is the is the area of the base and h is the height of the cone. area of the base and h is the height of the pyramid. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 3 105.8 mm ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary. 12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. 10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 482.1 m3

ANSWER: 605 in3 13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid. 11. The base is a hexagon, so we need to make a right tri SOLUTION: determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl The volume of a pyramid is , where B is the 90° triangle. area of the base and h is the height of the pyramid.

ANSWER: 3 105.8 mm

The apothem is .

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet ANSWER: SOLUTION: 482.1 m3 The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

13. SOLUTION:

The volume of a pyramid is , where B is the ANSWER: base and h is the height of the pyramid. 3 1376.7 ft The base is a hexagon, so we need to make a right tri 15. a triangular pyramid with a height of 4.8 centimeters determine the apothem. The interior angles of the he and a right triangle base with a leg 5 centimeters and 120°. The radius bisects the angle, so the right triangl hypotenuse 10.2 centimeters 90° triangle. SOLUTION: Find the height of the right triangle.

The apothem is .

ANSWER: The volume of a pyramid is , where B is the 3 233.8 cm area of the base and h is the height of the pyramid.

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. ANSWER: 3 35.6 cm

16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has a volume of 144 cubic centimeters. Find the height. SOLUTION: ANSWER: The base of the pyramid is a right triangle with a leg 1376.7 ft3 of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a 15. a triangular pyramid with a height of 4.8 centimeters of the right triangle and then find the area of the and a right triangle base with a leg 5 centimeters and triangle. hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2. The volume of a pyramid is , where B is the Replace V with 144 and B with 24 in the formula for area of the base and h is the height of the pyramid. the volume of a pyramid and solve for the height h.

ANSWER:

3 35.6 cm Therefore, the height of the triangular pyramid is 18 cm. 16. A triangular pyramid with a right triangle base with a leg 8 centimeters and hypotenuse 10 centimeters has ANSWER:

a volume of 144 cubic centimeters. Find the height. 18 cm SOLUTION: Find the volume of each cone. Round to the The base of the pyramid is a right triangle with a leg nearest tenth. of 8 centimeters and a hypotenuse of 10 centimeters. Use the Pythagorean Theorem to find the other leg a of the right triangle and then find the area of the triangle. 17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

The length of the other leg of the right triangle is 6 cm.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 3 235.6 in So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the height of the triangular pyramid is 18

cm.

ANSWER: 18 cm

Find the volume of each cone. Round to the nearest tenth. Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 3 17. 134.8 cm SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. 19.

Since the diameter of this cone is 10 inches, the SOLUTION: radius is or 5 inches. The height of the cone is 9 inches.

Use a trigonometric ratio to find the height h of the cone. 3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3 The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and Therefore, the volume of the cone is about 1473.1 the height is 7.3 centimeters. 3 cm . ANSWER: 1473.1 cm3

Therefore, the volume of the cone is about 134.8 3 20. cm . SOLUTION: ANSWER: 134.8 cm3

Use trigonometric ratios to find the height h and the 19. radius r of the cone.

SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the Use a trigonometric ratio to find the height h of the cone.

cone.

The volume of a circular cone is , where

r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters. 3 Therefore, the volume of the cone is about 2.8 ft . ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION: Therefore, the volume of the cone is about 1473.1 3 cm . The volume of a circular cone is , where

ANSWER: r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, 3 1473.1 cm the radius is or 8 inches.

20. SOLUTION:

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

Use trigonometric ratios to find the height h and the 22. a right cone with a slant height of 5.6 centimeters radius r of the cone. and a radius of 1 centimeter

SOLUTION: The cone has a radius r of 1 centimeter and a slant height of 5.6 centimeters. Use the Pythagorean Theorem to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft . ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches

SOLUTION: 3 Therefore, the volume of the cone is about 5.8 cm . The volume of a circular cone is , where ANSWER: r is the radius of the base and h is the height of the 3 cone. Since the diameter of this cone is 16 inches, 5.8 cm

the radius is or 8 inches. 23. SNACKS Approximately how many cubic centimeters of roasted peanuts will completely fill a paper cone that is 14 centimeters high and has a base diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where Therefore, the volume of the cone is about 1072.3 r is the radius of the base and h is the height of the 3 in . cone. Since the diameter of the cone is 8 ANSWER: centimeters, the radius is or 4 centimeters. The 3 height of the cone is 14 centimeters. 1072.3 in

ANSWER: 234.6 cm3

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

3 Therefore, the volume of the cone is about 5.8 cm . ANSWER: ANSWER: 3 42,000,000 ft 5.8 cm3 25. GARDENING The greenhouse is a regular 23. SNACKS Approximately how many cubic octagonal pyramid with a height of 5 feet. The base centimeters of roasted peanuts will completely fill a has side lengths of 2 feet. What is the volume of the paper cone that is 14 centimeters high and has a base greenhouse? diameter of 8 centimeters? Round to the nearest tenth. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of the cone is 8 SOLUTION: centimeters, the radius is or 4 centimeters. The height of the cone is 14 centimeters. The volume of a pyramid is , where B is the

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

24. CCSS MODELING The Pyramid Arena in

The volume of a pyramid is , where B is the The height of this pyramid is 5 feet. area of the base and h is the height of the pyramid.

ANSWER: 3 42,000,000 ft Therefore, the volume of the greenhouse is about 32.2 ft3. 25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base ANSWER: 3 has side lengths of 2 feet. What is the volume of the 32.2 ft greenhouse? Find the volume of each solid. Round to the nearest tenth.

26. SOLUTION: SOLUTION: The volume of a pyramid is , where B is the Volume of the solid given = Volume of the small cone + Volume of the large cone area of the base and h is the height of the pyramid. The base of the pyramid is a regular octagon with sides of 2 feet. A central angle of the octagon is or 45°, so the angle formed in the triangle below is 22.5°.

ANSWER: Use a trigonometric ratio to find the apothem a. 3 471.2 in

The height of this pyramid is 5 feet. 27.

SOLUTION:

Therefore, the volume of the greenhouse is about 32.2 ft3. ANSWER: 3 ANSWER: 3190.6 m 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

28. SOLUTION: 26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 7698.5 cm3

27. SOLUTION: SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

The volume of the ceiling is ANSWER: 3 3190.6 m

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic 28. foot, so the size of the heating unit Sam should buy is SOLUTION: 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

3 7698.5 cm Determine the volume of the model. Explain why knowing the volume is helpful in this situation. 29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she SOLUTION: needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction The volume of a pyramid is , where B is the with good insulation, there should be 2 BTUs per area of the base and h is the height of the pyramid. cubic foot. What size unit does Sam need to purchase?

It tells Marta how much clay is needed to make the model. SOLUTION: The building can be broken down into the rectangular ANSWER: base and the pyramid ceiling. The volume of the base 2 in3; It tells Marta how much clay is needed to is make the model.

31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. Describe how each change affects the volume of the cone.

a. The height is doubled. The volume of the ceiling is b. The radius is doubled. c. Both the radius and the height are doubled. SOLUTION: Find the volume of the original cone. Then alter the values.

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs a. Double h.

30. SCIENCE Marta is studying crystals that grow on rock formations.For a project, she is making a clay model of a crystal with a shape that is a composite of two congruent rectangular pyramids. The base of each pyramid will be 1 by 1.5 inches, and the total height will be 4 inches. The volume is doubled. Determine the volume of the model. Explain why knowing the volume is helpful in this situation. b. Double r. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

It tells Marta how much clay is needed to make the model.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model. volume is multiplied by 23 or 8. 31. CHANGING DIMENSIONS A cone has a radius of 4 centimeters and a height of 9 centimeters. ANSWER: Describe how each change affects the volume of the a. The volume is doubled. cone. 2 b. The volume is multiplied by 2 or 4. a. The height is doubled. 3 b. The radius is doubled. c. The volume is multiplied by 2 or 8. c. Both the radius and the height are doubled. Find each measure. Round to the nearest tenth SOLUTION: if necessary. Find the volume of the original cone. Then alter the 32. A square pyramid has a volume of 862.5 cubic values. centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

a. Double h.

The volume is doubled.

b. Double r.

The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the 2 The volume is multiplied by 2 or 4. height is 12 inches. What is the diameter? SOLUTION: c. Double r and h. The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

3 volume is multiplied by 2 or 8. ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. The diameter is 2(7) or 14 inches. SOLUTION: ANSWER: The volume of a pyramid is , where B is the 14 in. area of the base and h is the height of the pyramid. 34. The lateral area of a cone is 71.6 square millimeters Let s be the side length of the base. and the slant height is 6 millimeters. What is the volume of the cone?

SOLUTION:

The lateral area of a cone is , where r is the radius and is the slant height of the cone. The side length of the base is 15 cm. Replace L with 71.6 and with 6, then solve for the radius r. ANSWER:

15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter?

SOLUTION: So, the radius is about 3.8 millimeters. The volume of a circular cone is , or

Use the Pythagorean Theorem to find the height of , where B is the area of the base, h is the the cone. height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

The diameter is 2(7) or 14 inches.

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the The lateral area of a cone is , where r is two pyramids that you drew? Explain. the radius and is the slant height of the cone. c. ANALYTICAL Explain how multiplying the base Replace L with 71.6 and with 6, then solve for the area and/or the height of the pyramid by 5 affects the radius r. volume of the pyramid.

SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer: So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. b. The volumes are the same. The volume of a pyramid equals one third times the base area times the height. So, if the base areas of two pyramids are equal and their heights are equal, then their volumes are equal.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. c. If the base area is multiplied by 5, the volume is b. VERBAL What is true about the volumes of the multiplied by 5. If the height is multiplied by 5, the two pyramids that you drew? Explain. volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied c. ANALYTICAL Explain how multiplying the base by 5 · 5 or 25. area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

The volume of a cone with a radius r and height h is . The volume of a prism with a height of h is where B is the area of the base of the prism. Set the volumes equal.

ANSWER: a. Sample answer:

or 10h cubic units.

ANSWER: Sometimes; the statement is true if the base area of b. The volumes are the same. The volume of a the cone is 3 times as great as the base area of the pyramid equals one third times the base area times prism. For example, if the base of the prism has an the height. So, if the base areas of two pyramids are area of 10 square units, then its volume is 10h cubic equal and their heights are equal, then their volumes units. So, the cone must have a base area of 30 are equal. c. If the base area is multiplied by 5, the volume is square units so that its volume is or 10h multiplied by 5. If the height is multiplied by 5, the cubic units. volume is multiplied by 5. If both the base area and the height are multiplied by 5, the volume is multiplied 37. ERROR ANALYSIS Alexandra and Cornelio are by 5 · 5 or 25. calculating the volume of the cone below. Is either of them correct? Explain your answer. 36. CCSS ARGUMENTS Determine whether the following statement is sometimes, always, or never true. Justify your reasoning. The volume of a cone with radius r and height h equals the volume of a prism with height h.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height.

REASONING 38. A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that The volumes will only be equal when the radius of has the same radius and height as the cone? Explain your reasoning. the cone is equal to or when . SOLUTION: Therefore, the statement is true sometimes if the 3 base area of the cone is 3 times as great as the base 1704 cm ; The formula for the volume of a cylinder area of the prism. For example, if the base of the is V= Bh, while the formula for the volume of a cone prism has an area of 10 square units, then its volume is V = Bh. The volume of a cylinder is three times is 10h cubic units. So, the cone must have a base as much as the volume of a cone with the same area of 30 square units so that its volume is radius and height.

or 10h cubic units. ANSWER: 1704 cm3; The volume of a cylinder is three times as ANSWER: much as the volume of a cone with the same radius Sometimes; the statement is true if the base area of and height. the cone is 3 times as great as the base area of the prism. For example, if the base of the prism has an 39. OPEN ENDED Give an example of a pyramid and area of 10 square units, then its volume is 10h cubic a prism that have the same base and the same units. So, the cone must have a base area of 30 volume. Explain your reasoning. square units so that its volume is or 10h SOLUTION: cubic units. The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of 37. ERROR ANALYSIS Alexandra and Cornelio are that. So, if a pyramid and prism have the same base, calculating the volume of the cone below. Is either of then in order to have the same volume, the height of them correct? Explain your answer. the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism. SOLUTION: The slant height is used for surface area, but the Sample answer: height is used for volume. For this cone, the slant A square pyramid with a base area of 16 and a height of 13 is provided, and we need to calculate the height of 12, a prism with a square base area of 16 height before we can calculate the volume. and a height of 4.

ANSWER: Alexandra incorrectly used the slant height. Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base ANSWER: area of 16 and a height of 4; if a pyramid and prism Cornelio; Alexandra incorrectly used the slant height. have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as 38. REASONING A cone has a volume of 568 cubic great as the height of the prism. centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain 40. WRITING IN MATH Compare and contrast your reasoning. finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: 3 SOLUTION: 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone To find the volume of each solid, you must know the area of the base and the height. The volume of a is V = Bh. The volume of a cylinder is three times pyramid is one third the volume of a prism that has as much as the volume of a cone with the same the same height and base area. The volume of a radius and height. cone is one third the volume of a cylinder that has the same height and base area. ANSWER: 1704 cm3; The volume of a cylinder is three times as ANSWER: much as the volume of a cone with the same radius To find the volume of each solid, you must know the and height. area of the base and the height. The volume of a pyramid is one third the volume of a prism that has 39. OPEN ENDED Give an example of a pyramid and the same height and base area. The volume of a a prism that have the same base and the same cone is one third the volume of a cylinder that has the volume. Explain your reasoning. same height and base area.

SOLUTION: 41. A conical sand toy has the dimensions as shown The formula for volume of a prism is V = Bh and the below. How many cubic centimeters of sand will it formula for the volume of a pyramid is one-third of hold when it is filled to the top? that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

A 12π B 15π

D Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the SOLUTION: height of the prism. Use the Pythagorean Theorem to find the radius r of the cone. Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. So, the radius of the cone is 3 centimeters. SOLUTION: To find the volume of each solid, you must know the The volume of a circular cone is , where area of the base and the height. The volume of a r is the radius of the base and h is the height of the pyramid is one third the volume of a prism that has cone. the same height and base area. The volume of a

cone is one third the volume of a cylinder that has the same height and base area.

ANSWER: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a Therefore, the correct choice is A. cone is one third the volume of a cylinder that has the same height and base area. ANSWER: A 41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it 42. SHORT RESPONSE Brooke is buying a tent that hold when it is filled to the top? is in the shape of a rectangular pyramid. The base is 6 feet by 8 feet. If the tent holds 88 cubic feet of air, how tall is the tent’s center pole? SOLUTION:

A 12π The volume of a pyramid is , where B is the B 15π area of the base and h is the height of the pyramid. C

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

So, the radius of the cone is 3 centimeters.

F The volume of a circular cone is , where

r is the radius of the base and h is the height of the G cone.

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Therefore, the correct choice is A. Number of possible outcomes : 25 Favorable outcomes: {5 orange} ANSWER: Number of favorable outcomes: 5 A

The volume of a pyramid is , where B is the So, the correct choice is F. area of the base and h is the height of the pyramid. ANSWER: F

44. SAT/ACT For all

A –8 B x – 4 C D E ANSWER: 5.5 ft SOLUTION: 43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange? So, the correct choice is E.

ANSWER: E

Find the volume of each prism. F

H 45. SOLUTION: J The volume of a prism is , where B is the area of the base and h is the height of the prism. The SOLUTION: base of this prism is a rectangle with a length of 14 Possible outcomes: {6 red, 4 blue, 5 orange, 10 inches and a width of 12 inches. The height h of the green} prism is 6 inches. Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 3 So, the correct choice is F. 1008 in

ANSWER: F

44. SAT/ACT For all 46. A –8 SOLUTION: B x – 4 The volume of a prism is , where B is the

C area of the base and h is the height of the prism. The D base of this prism is an isosceles triangle with a base of 10 feet and two legs of 13 feet. The height h will E bisect the base. Use the Pythagorean Theorem to find the height of the triangle. SOLUTION:

So, the correct choice is E.

ANSWER: E Find the volume of each prism.

So, the height of the triangle is 12 feet. Find the area of the triangle. 45. SOLUTION: The volume of a prism is , where B is the area of the base and h is the height of the prism. The base of this prism is a rectangle with a length of 14 inches and a width of 12 inches. The height h of the prism is 6 inches. So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

3 Therefore, the volume of the prism is 1008 in . Therefore, the volume of the prism is 1140 ft3. ANSWER: 1008 in3 ANSWER: 3 1140 ft

46. SOLUTION: 47. The volume of a prism is , where B is the SOLUTION: area of the base and h is the height of the prism. The base of this prism is an isosceles triangle with a base The volume of a prism is , where B is the of 10 feet and two legs of 13 feet. The height h will area of the base and h is the height of the prism. The bisect the base. Use the Pythagorean Theorem to base of this prism is a rectangle with a length of 79.4 find the height of the triangle. meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6

3 m . ANSWER: 3 426,437.6 m

48. FARMING The picture shows a combination So, the height of the triangle is 12 feet. Find the area hopper cone and bin used by farmers to store grain of the triangle. after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the dimensions in the diagram to find the entire surface area of the bin with a conical top and bottom. Write the exact answer and the answer rounded to the nearest square foot. Refer to the photo on Page 863. SOLUTION: 2 So, the area of the base B is 60 ft . To find the entire surface area of the bin, find the surface area of the conical top and bottom and find The height h of the prism is 19 feet. the surface area of the cylinder and add them. The formula for finding the surface area of a cone is , where is r is the radius and l is the slant height of the cone.

Find the slant height of the conical top.

3 Therefore, the volume of the prism is 1140 ft . ANSWER: 3 1140 ft

47.

SOLUTION: Find the slant height of the conical bottom. The volume of a prism is , where B is the The height of the conical bottom is 28 – (5 + 12 + 2) area of the base and h is the height of the prism. The or 9 ft. base of this prism is a rectangle with a length of 79.4 meters and a width of 52.5 meters. The height of the prism is 102.3 meters.

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: The formula for finding the surface area of a cylinder 3 426,437.6 m is , where is r is the radius and h is the slant height of the cylinder. 48. FARMING The picture shows a combination hopper cone and bin used by farmers to store grain after harvest. The cone at the bottom of the bin allows the grain to be emptied more easily. Use the Surface area of the bin = Surface area of the dimensions in the diagram to find the entire surface cylinder + Surface area of the conical top + surface area of the bin with a conical top and bottom. Write area of the conical bottom. the exact answer and the answer rounded to the

nearest square foot. Refer to the photo on Page 863. SOLUTION: To find the entire surface area of the bin, find the surface area of the conical top and bottom and find

the surface area of the cylinder and add them. ANSWER: The formula for finding the surface area of a cone is

, where is r is the radius and l is the slant height of the cone. Find the area of each shaded region. Polygons Find the slant height of the conical top. in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft. ANSWER:

30.4 cm2

50.

SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

A regular hexagon has 6 sides, so the measure of the Surface area of the bin = Surface area of the interior angle is . The apothem bisects the cylinder + Surface area of the conical top + surface angle, so we use 30° when using trig to find the area of the conical bottom. length of the apothem.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

Now find the area of the hexagon.

ANSWER: 30.4 cm2

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon ANSWER: 2 54.4 in

A regular hexagon has 6 sides, so the measure of the 51. interior angle is . The apothem bisects the SOLUTION: angle, so we use 30° when using trig to find the The shaded area is the area of the equilateral triangle length of the apothem. less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Now find the area of the hexagon.

Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find the area of the equilateral triangle. ANSWER: 2 54.4 in

So, the area of the equilateral triangle is about 67.3 51. square feet. SOLUTION: The shaded area is the area of the equilateral triangle Subtract to find the area of shaded region. less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 12-5So, Volumes the area of of Pyramids the circle andis about Cones 40.7 square feet. 2 26.6 ft Next, find the area of the equilateral triangle.

52. SOLUTION: The area of the shaded region is the area of the circumscribed circle minus the area of the equilateral The equilateral triangle can be divided into three triangle plus the area of the inscribed circle. Let b isosceles triangles with central angles of or represent the length of each side of the equilateral 120°, so the angle in the triangle created by the triangle and h represent the radius of the inscribed height of the isosceles triangle is 60° and the base is circle. half the length b of the side of the equilateral triangle.

Divide the equilateral triangle into three isosceles Use a trigonometric ratio to find the value of b. triangles with central angles of or 120°. The

angle in the triangle formed by the height.h of the isosceles triangle is 60° and the base is half the length of the side of the equilateral triangle as shown below.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

Use trigonometric ratios to find the values of b and h.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find 26.6 ft2. the area of the shaded region. ANSWER: 2 A(shaded region) = A(circumscribed circle) - A 26.6 ft (equilateral triangle) + A(inscribed circle)

eSolutions Manual - Powered by Cognero Page 23 52.

SOLUTION: Therefore, the area of the shaded region is about The area of the shaded region is the area of the 168.2 mm2. circumscribed circle minus the area of the equilateral triangle plus the area of the inscribed circle. Let b represent the length of each side of the equilateral ANSWER: triangle and h represent the radius of the inscribed 2 circle. 168.2 mm

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region.

A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm Find the volume of each pyramid.

1. SOLUTION:

ANSWER: 3 75 in

2. SOLUTION:

ANSWER: 132 cm3

3. a rectangular pyramid with a height of 5.2 meters and a base 8 meters by 4.5 meters SOLUTION:

ANSWER: 3 62.4 m

4. a square pyramid with a height of 14 meters and a base with 8-meter side lengths SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a square with sides of 8 meters. The height of the pyramid is 14 meters.

ANSWER: 298.7 m3

Find the volume of each cone. Round to the nearest tenth.

5. SOLUTION:

The volume of a circular cone is , or

ANSWER: 3 51.3 in

6. SOLUTION:

Use trigonometry to find the radius r.

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The height of the cone is 11.5 centimeters.

ANSWER: 168.1 cm3

7. an oblique cone with a height of 10.5 millimeters and a radius of 1.6 millimeters SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. The radius of this cone is 1.6 millimeters and the height is 10.5 millimeters.

ANSWER: 3 28.1 mm

8. a cone with a slant height of 25 meters and a radius of 15 meters SOLUTION:

Use the Pythagorean Theorem to find the height h of the cone. Then find its volume.

So, the height of the cone is 20 meters.

ANSWER: 4712.4 m3

The volume of a circular cone is , where B is the area of the base and h is the height of the cone. For this cone, the area of the base is 15,400 square feet and the height is 100 feet.

ANSWER: 3 513,333.3 ft

CCSS SENSE-MAKING Find the volume of each pyramid. Round to the nearest tenth if necessary.

10. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 605 in3

11. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 105.8 mm

12. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 482.1 m3

13. SOLUTION:

The volume of a pyramid is , where B is the base and h is the height of the pyramid.

The base is a hexagon, so we need to make a right tri determine the apothem. The interior angles of the he 120°. The radius bisects the angle, so the right triangl 90° triangle.

The apothem is .

ANSWER: 3 233.8 cm

14. a pentagonal pyramid with a base area of 590 square feet and an altitude of 7 feet SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 1376.7 ft3

15. a triangular pyramid with a height of 4.8 centimeters and a right triangle base with a leg 5 centimeters and hypotenuse 10.2 centimeters SOLUTION: Find the height of the right triangle.

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 35.6 cm

The length of the other leg of the right triangle is 6 cm.

So, the area of the base B is 24 cm2.

Replace V with 144 and B with 24 in the formula for the volume of a pyramid and solve for the height h.

Therefore, the height of the triangular pyramid is 18 cm.

ANSWER: 18 cm Find the volume of each cone. Round to the nearest tenth.

17. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Since the diameter of this cone is 10 inches, the radius is or 5 inches. The height of the cone is 9 inches.

3 Therefore, the volume of the cone is about 235.6 in .

ANSWER: 235.6 in3

18. SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 4.2 centimeters and the height is 7.3 centimeters.

Therefore, the volume of the cone is about 134.8 3 cm .

ANSWER: 134.8 cm3

19. SOLUTION:

Use a trigonometric ratio to find the height h of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. The radius of this cone is 8 centimeters.

Therefore, the volume of the cone is about 1473.1 3 cm .

ANSWER: 1473.1 cm3

20. SOLUTION:

Use trigonometric ratios to find the height h and the radius r of the cone.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

3 Therefore, the volume of the cone is about 2.8 ft .

ANSWER: 2.8 ft3

21. an oblique cone with a diameter of 16 inches and an altitude of 16 inches SOLUTION:

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone. Since the diameter of this cone is 16 inches, the radius is or 8 inches.

Therefore, the volume of the cone is about 1072.3 3 in .

ANSWER: 1072.3 in3

3 Therefore, the volume of the cone is about 5.8 cm .

ANSWER: 5.8 cm3

3 Therefore, the paper cone will hold about 234.6 cm of roasted peanuts.

ANSWER: 234.6 cm3

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 3 42,000,000 ft

25. GARDENING The greenhouse is a regular octagonal pyramid with a height of 5 feet. The base has side lengths of 2 feet. What is the volume of the greenhouse?

SOLUTION:

Use a trigonometric ratio to find the apothem a.

The height of this pyramid is 5 feet.

Therefore, the volume of the greenhouse is about 32.2 ft3.

ANSWER: 3 32.2 ft

Find the volume of each solid. Round to the nearest tenth.

26. SOLUTION: Volume of the solid given = Volume of the small cone + Volume of the large cone

ANSWER: 471.2 in3

27. SOLUTION:

ANSWER: 3 3190.6 m

28. SOLUTION:

ANSWER: 7698.5 cm3

29. HEATING Sam is building an art studio in her backyard. To buy a heating unit for the space, she needs to determine the BTUs (British Thermal Units) required to heat the building. For new construction with good insulation, there should be 2 BTUs per cubic foot. What size unit does Sam need to purchase?

SOLUTION: The building can be broken down into the rectangular base and the pyramid ceiling. The volume of the base is

The volume of the ceiling is

The total volume is therefore 5000 + 1666.67 = 3 6666.67 ft . Two BTU's are needed for every cubic foot, so the size of the heating unit Sam should buy is 6666.67 × 2 = 13,333 BTUs.

ANSWER: 13,333 BTUs

Determine the volume of the model. Explain why knowing the volume is helpful in this situation. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

It tells Marta how much clay is needed to make the model.

ANSWER: 2 in3; It tells Marta how much clay is needed to make the model.

a. Double h.

The volume is doubled.

b. Double r.

2 The volume is multiplied by 2 or 4.

c. Double r and h.

volume is multiplied by 23 or 8.

ANSWER: a. The volume is doubled. 2 b. The volume is multiplied by 2 or 4. c. The volume is multiplied by 23 or 8.

Find each measure. Round to the nearest tenth if necessary. 32. A square pyramid has a volume of 862.5 cubic centimeters and a height of 11.5 centimeters. Find the side length of the base. SOLUTION:

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid. Let s be the side length of the base.

The side length of the base is 15 cm.

ANSWER: 15 cm

33. The volume of a cone is 196π cubic inches and the height is 12 inches. What is the diameter? SOLUTION:

The volume of a circular cone is , or

, where B is the area of the base, h is the height of the cone, and r is the radius of the base. Since the diameter is 8 centimeters, the radius is 4 centimeters.

The diameter is 2(7) or 14 inches.

ANSWER: 14 in. 34. The lateral area of a cone is 71.6 square millimeters and the slant height is 6 millimeters. What is the volume of the cone? SOLUTION:

The lateral area of a cone is , where r is the radius and is the slant height of the cone. Replace L with 71.6 and with 6, then solve for the radius r.

So, the radius is about 3.8 millimeters.

Use the Pythagorean Theorem to find the height of the cone.

So, the height of the cone is about 4.64 millimeters.

The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the volume of the cone is about 70.2 3 mm .

ANSWER: 70.2 mm3

35. MULTIPLE REPRESENTATIONS In this problem, you will investigate rectangular pyramids. a. GEOMETRIC Draw two pyramids with different bases that have a height of 10 centimeters and a base area of 24 square centimeters. b. VERBAL What is true about the volumes of the two pyramids that you drew? Explain. c. ANALYTICAL Explain how multiplying the base area and/or the height of the pyramid by 5 affects the volume of the pyramid. SOLUTION: a. Use rectangular bases and pick values that multiply to make 24.

Sample answer:

ANSWER: a. Sample answer:

or 10h cubic units.

37. ERROR ANALYSIS Alexandra and Cornelio are calculating the volume of the cone below. Is either of them correct? Explain your answer.

Alexandra incorrectly used the slant height.

ANSWER: Cornelio; Alexandra incorrectly used the slant height.

38. REASONING A cone has a volume of 568 cubic centimeters. What is the volume of a cylinder that has the same radius and height as the cone? Explain your reasoning. SOLUTION: 3 1704 cm ; The formula for the volume of a cylinder is V= Bh, while the formula for the volume of a cone is V = Bh. The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

ANSWER: 1704 cm3; The volume of a cylinder is three times as much as the volume of a cone with the same radius and height.

39. OPEN ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. SOLUTION: The formula for volume of a prism is V = Bh and the formula for the volume of a pyramid is one-third of that. So, if a pyramid and prism have the same base, then in order to have the same volume, the height of the pyramid must be 3 times as great as the height of the prism.

Set the base areas of the prism and pyramid, and make the height of the pyramid equal to 3 times the height of the prism.

Sample answer: A square pyramid with a base area of 16 and a height of 12, a prism with a square base area of 16 and a height of 4.

40. WRITING IN MATH Compare and contrast finding volumes of pyramids and cones with finding volumes of prisms and cylinders. SOLUTION: To find the volume of each solid, you must know the area of the base and the height. The volume of a pyramid is one third the volume of a prism that has the same height and base area. The volume of a cone is one third the volume of a cylinder that has the same height and base area.

41. A conical sand toy has the dimensions as shown below. How many cubic centimeters of sand will it hold when it is filled to the top?

A 12π B 15π

SOLUTION: Use the Pythagorean Theorem to find the radius r of the cone.

So, the radius of the cone is 3 centimeters. The volume of a circular cone is , where r is the radius of the base and h is the height of the cone.

Therefore, the correct choice is A.

ANSWER: A

The volume of a pyramid is , where B is the area of the base and h is the height of the pyramid.

ANSWER: 5.5 ft

43. PROBABILITY A spinner has sections colored red, blue, orange, and green. The table below shows the results of several spins. What is the experimental probability of the spinner landing on orange?

SOLUTION: Possible outcomes: {6 red, 4 blue, 5 orange, 10 green} Number of possible outcomes : 25 Favorable outcomes: {5 orange} Number of favorable outcomes: 5

So, the correct choice is F.

ANSWER: F

44. SAT/ACT For all

A –8 B x – 4 C D E

SOLUTION:

So, the correct choice is E.

ANSWER: E Find the volume of each prism.

3 Therefore, the volume of the prism is 1008 in .

ANSWER: 1008 in3

So, the height of the triangle is 12 feet. Find the area of the triangle.

So, the area of the base B is 60 ft2.

The height h of the prism is 19 feet.

Therefore, the volume of the prism is 1140 ft3.

ANSWER: 3 1140 ft

Therefore, the volume of the prism is about 426,437.6 m3.

ANSWER: 3 426,437.6 m

Find the slant height of the conical top.

Find the slant height of the conical bottom. The height of the conical bottom is 28 – (5 + 12 + 2) or 9 ft.

The formula for finding the surface area of a cylinder is , where is r is the radius and h is the slant height of the cylinder.

Surface area of the bin = Surface area of the cylinder + Surface area of the conical top + surface area of the conical bottom.

ANSWER:

Find the area of each shaded region. Polygons in 50 - 52 are regular.

49. SOLUTION: Area of the shaded region = Area of the rectangle – Area of the circle Area of the rectangle = 10(5) = 50

ANSWER: 30.4 cm2

50. SOLUTION: Area of the shaded region = Area of the circle – Area of the hexagon

A regular hexagon has 6 sides, so the measure of the interior angle is . The apothem bisects the angle, so we use 30° when using trig to find the length of the apothem.

Now find the area of the hexagon.

ANSWER: 2 54.4 in

51. SOLUTION: The shaded area is the area of the equilateral triangle less the area of the inscribed circle.

Find the area of the circle with a radius of 3.6 feet.

So, the area of the circle is about 40.7 square feet.

Next, find the area of the equilateral triangle.

Use a trigonometric ratio to find the value of b.

Now use the area of the isosceles triangles to find the area of the equilateral triangle.

So, the area of the equilateral triangle is about 67.3 square feet.

Subtract to find the area of shaded region.

Therefore, the area of the shaded region is about 26.6 ft2.

ANSWER: 2 26.6 ft

Use trigonometric ratios to find the values of b and h.

The area of the equilateral triangle will equal three times the area of any of the isosceles triangles. Find the area of the shaded region. 12-5 Volumes of Pyramids and Cones A(shaded region) = A(circumscribed circle) - A (equilateral triangle) + A(inscribed circle)

Therefore, the area of the shaded region is about 168.2 mm2.

ANSWER: 2 168.2 mm